11.3. ERGODIC MARKOV CHAINS 441
12
3
65 4
78 9
Figure 11.4: The maze problem.
To find the fixed probability vector for this matrix, we would have to solve ten
equations in nine unknowns. However, it would seem reasonable that the times
spent in each compartment should, in the long run, be proportional to the number
of entries to each compartment. Thus, we try the vector whose jth component is
the number of entries to the jth compartment:
x = (2 3 2 3 4 3 2 3 2) .
It is easy to check that this vector is indeed a fixed vector so that the unique
probability vector is this vector normalized to have sum 1:
w = ( 1 1 1 1 1 1 1 1 1 ) .
12 8 12 8 6 8 12 8 12
2
Example 11.23 (Example 11.8 continued) We recall the Ehrenfest urn model of
Example 11.8. The transition matrix for this chain is as follows:
01234
0 .000 1.000 .000 .000 .000
1 .250 .000 .750 .000 .000
P= 2 .000 .500 .000 .500 .000 .
3 .000 .000 .750 .000 .250
4 .000 .000 .000 1.000 .000
If we run the program FixedVector for this chain, we obtain the vector
01234
w = .0625 .2500 .3750 .2500 .0625 .
By Theorem 11.12, we can interpret these values for wi as the proportion of times
the process is in each of the states in the long run. For example, the proportion of
442 CHAPTER 11. MARKOV CHAINS
times in state 0 is .0625 and the proportion of times in state 1 is .375. The astute
reader will note that these numbers are the binomial distribution 1/16, 4/16, 6/16,
4/16, 1/16. We could have guessed this answer as follows: If we consider a particular
ball, it simply moves randomly back and forth between the two urns. This suggests
that the equilibrium state should be just as if we randomly distributed the four
balls in the two urns. If we did this, the probability that there would be exactly
j balls in one urn would be given by the binomial distribution b(n, p, j) with n = 4
and p = 1/2. 2
Exercises
1 Which of the following matrices are transition matrices for regular Markov
chains?
(a) P = .5 .5 .
.5 .5
(b) P = .5 .5 .
1 0
1/3 0 2/3
(c) P = 0 1 0 .
0 1/5 4/5
(d) P = 01 .
10
1/2 1/2 0
(e) P = 0 1/2 1/2 .
1/3 1/3 1/3
2 Consider the Markov chain with transition matrix
1/2 1/3 1/6
P = 3/4 0 1/4 .
010
(a) Show that this is a regular Markov chain.
(b) The process is started in state 1; find the probability that it is in state 3
after two steps.
(c) Find the limiting probability vector w.
3 Consider the Markov chain with general 2 × 2 transition matrix
P= 1−a a .
b 1−b
(a) Under what conditions is P absorbing?
(b) Under what conditions is P ergodic but not regular?
(c) Under what conditions is P regular?
11.3. ERGODIC MARKOV CHAINS 443
4 Find the fixed probability vector w for the matrices in Exercise 3 that are
ergodic.
5 Find the fixed probability vector w for each of the following regular matrices.
(a) P = .75 .25 .
.5 .5
(b) P = .9 .1 .
.1 .9
3/4 1/4 0
(c) P = 0 2/3 1/3 .
1/4 1/4 1/2
6 Consider the Markov chain with transition matrix in Exercise 3, with a = b =
1. Show that this chain is ergodic but not regular. Find the fixed probability
vector and interpret it. Show that Pn does not tend to a limit, but that
An = I + P + P2 + · · · + Pn
n+1
does.
7 Consider the Markov chain with transition matrix of Exercise 3, with a = 0
and b = 1/2. Compute directly the unique fixed probability vector, and use
your result to prove that the chain is not ergodic.
8 Show that the matrix 1 0 0
P = 1/4 1/2 1/4
001
has more than one fixed probability vector. Find the matrix that Pn ap-
proaches as n → ∞, and verify that it is not a matrix all of whose rows are
the same.
9 Prove that, if a 3-by-3 transition matrix has the property that its column sums
are 1, then (1/3, 1/3, 1/3) is a fixed probability vector. State a similar result
for n-by-n transition matrices. Interpret these results for ergodic chains.
10 Is the Markov chain in Example 11.10 ergodic?
11 Is the Markov chain in Example 11.11 ergodic?
12 Consider Example 11.13 (Drunkard’s Walk). Assume that if the walker reaches
state 0, he turns around and returns to state 1 on the next step and, simi-
larly, if he reaches 4 he returns on the next step to state 3. Is this new chain
ergodic? Is it regular?
13 For Example 11.4 when P is ergodic, what is the proportion of people who
are told that the President will run? Interpret the fact that this proportion
is independent of the starting state.
444 CHAPTER 11. MARKOV CHAINS
14 Consider an independent trials process to be a Markov chain whose states are
the possible outcomes of the individual trials. What is its fixed probability
vector? Is the chain always regular? Illustrate this for Example 11.5.
15 Show that Example 11.8 is an ergodic chain, but not a regular chain. Show
that its fixed probability vector w is a binomial distribution.
16 Show that Example 11.9 is regular and find the limiting vector.
17 Toss a fair die repeatedly. Let Sn denote the total of the outcomes through
the nth toss. Show that there is a limiting value for the proportion of the first
n values of Sn that are divisible by 7, and compute the value for this limit.
Hint: The desired limit is an equilibrium probability vector for an appropriate
seven state Markov chain.
18 Let P be the transition matrix of a regular Markov chain. Assume that there
are r states and let N (r) be the smallest integer n such that P is regular if
and only if PN(r) has no zero entries. Find a finite upper bound for N (r).
See if you can determine N (3) exactly.
*19 Define f (r) to be the smallest integer n such that for all regular Markov chains
with r states, the nth power of the transition matrix has all entries positive.
It has been shown,14 that f (r) = r2 − 2r + 2.
(a) Define the transition matrix of an r-state Markov chain as follows: For
states si, with i = 1, 2, . . . , r−2, P(i, i+1) = 1, P(r−1, r) = P(r−1, 1) =
1/2, and P(r, 1) = 1. Show that this is a regular Markov chain.
(b) For r = 3, verify that the fifth power is the first power that has no zeros.
(c) Show that, for general r, the smallest n such that Pn has all entries
positive is n = f (r).
20 A discrete time queueing system of capacity n consists of the person being
served and those waiting to be served. The queue length x is observed each
second. If 0 < x < n, then with probability p, the queue size is increased by
one by an arrival and, inependently, with probability r, it is decreased by one
because the person being served finishes service. If x = 0, only an arrival (with
probability p) is possible. If x = n, an arrival will depart without waiting for
service, and so only the departure (with probability r) of the person being
served is possible. Form a Markov chain with states given by the number of
customers in the queue. Modify the program FixedVector so that you can
input n, p, and r, and the program will construct the transition matrix and
compute the fixed vector. The quantity s = p/r is called the traffic intensity.
Describe the differences in the fixed vectors according as s < 1, s = 1, or
s > 1.
14E. Seneta, Non-Negative Matrices: An Introduction to Theory and Applications, Wiley, New
York, 1973, pp. 52-54.
11.3. ERGODIC MARKOV CHAINS 445
21 Write a computer program to simulate the queue in Exercise 20. Have your
program keep track of the proportion of the time that the queue length is j for
j = 0, 1, . . . , n and the average queue length. Show that the behavior of the
queue length is very different depending upon whether the traffic intensity s
has the property s < 1, s = 1, or s > 1.
22 In the queueing problem of Exercise 20, let S be the total service time required
by a customer and T the time between arrivals of the customers.
(a) Show that P (S = j) = (1 − r)j−1r and P (T = j) = (1 − p)j−1p, for
j > 0.
(b) Show that E(S) = 1/r and E(T ) = 1/p.
(c) Interpret the conditions s < 1, s = 1 and s > 1 in terms of these expected
values.
23 In Exercise 20 the service time S has a geometric distribution with E(S) =
1/r. Assume that the service time is, instead, a constant time of t seconds.
Modify your computer program of Exercise 21 so that it simulates a constant
time service distribution. Compare the average queue length for the two
types of distributions when they have the same expected service time (i.e.,
take t = 1/r). Which distribution leads to the longer queues on the average?
24 A certain experiment is believed to be described by a two-state Markov chain
with the transition matrix P, where
.5 .5
P= p 1−p
and the parameter p is not known. When the experiment is performed many
times, the chain ends in state one approximately 20 percent of the time and in
state two approximately 80 percent of the time. Compute a sensible estimate
for the unknown parameter p and explain how you found it.
25 Prove that, in an r-state ergodic chain, it is possible to go from any state to
any other state in at most r − 1 steps.
26 Let P be the transition matrix of an r-state ergodic chain. Prove that, if the
diagonal entries pii are positive, then the chain is regular.
27 Prove that if P is the transition matrix of an ergodic chain, then (1/2)(I + P)
is the transition matrix of a regular chain. Hint: Use Exercise 26.
28 Prove that P and (1/2)(I + P) have the same fixed vectors.
29 In his book, Wahrscheinlichkeitsrechnung und Statistik,15 A. Engle proposes
an algorithm for finding the fixed vector for an ergodic Markov chain when
the transition probabilities are rational numbers. Here is his algorithm: For
15A. Engle, Wahrscheinlichkeitsrechnung und Statistik, vol. 2 (Stuttgart: Klett Verlag, 1976).
446 CHAPTER 11. MARKOV CHAINS
(4 2 4)
(5 2 3)
(8 2 4)
(7 3 4)
(8 4 4)
(8 3 5)
(8 4 8)
(10 4 6)
(12 4 8)
(12 5 7)
(12 6 8)
(13 5 8)
(16 6 8)
(15 6 9)
(16 6 12)
(17 7 10)
(20 8 12)
(20 8 12) .
Table 11.4: Distribution of chips.
each state i, let ai be the least common multiple of the denominators of the
non-zero entries in the ith row. Engle describes his algorithm in terms of mov-
ing chips around on the states—indeed, for small examples, he recommends
implementing the algorithm this way. Start by putting ai chips on state i for
all i. Then, at each state, redistribute the ai chips, sending aipij to state j.
The number of chips at state i after this redistribution need not be a multiple
of ai. For each state i, add just enough chips to bring the number of chips at
state i up to a multiple of ai. Then redistribute the chips in the same manner.
This process will eventually reach a point where the number of chips at each
state, after the redistribution, is the same as before redistribution. At this
point, we have found a fixed vector. Here is an example:
123
1 1/2 1/4 1/4
P = 2 1/2 0 1/2 .
3 1/2 1/4 1/4
We start with a = (4, 2, 4). The chips after successive redistributions are
shown in Table 11.4.
We find that a = (20, 8, 12) is a fixed vector.
(a) Write a computer program to implement this algorithm.
(b) Prove that the algorithm will stop. Hint: Let b be a vector with integer
components that is a fixed vector for P and such that each coordinate of
11.4. FUNDAMENTAL LIMIT THEOREM 447
the starting vector a is less than or equal to the corresponding component
of b. Show that, in the iteration, the components of the vectors are
always increasing, and always less than or equal to the corresponding
component of b.
30 (Coffman, Kaduta, and Shepp16) A computing center keeps information on a
tape in positions of unit length. During each time unit there is one request to
occupy a unit of tape. When this arrives the first free unit is used. Also, during
each second, each of the units that are occupied is vacated with probability p.
Simulate this process, starting with an empty tape. Estimate the expected
number of sites occupied for a given value of p. If p is small, can you choose the
tape long enough so that there is a small probability that a new job will have
to be turned away (i.e., that all the sites are occupied)? Form a Markov chain
with states the number of sites occupied. Modify the program FixedVector
to compute the fixed vector. Use this to check your conjecture by simulation.
*31 (Alternate proof of Theorem 11.8) Let P be the transition matrix of an ergodic
Markov chain. Let x be any column vector such that Px = x. Let M be the
maximum value of the components of x. Assume that xi = M . Show that if
pij > 0 then xj = M . Use this to prove that x must be a constant vector.
32 Let P be the transition matrix of an ergodic Markov chain. Let w be a fixed
probability vector (i.e., w is a row vector with wP = w). Show that if wi = 0
and pji > 0 then wj = 0. Use this to show that the fixed probability vector
for an ergodic chain cannot have any 0 entries.
33 Find a Markov chain that is neither absorbing or ergodic.
11.4 Fundamental Limit Theorem for Regular
Chains
The fundamental limit theorem for regular Markov chains states that if P is a
regular transition matrix then
lim Pn = W ,
n→∞
where W is a matrix with each row equal to the unique fixed probability row vector
w for P. In this section we shall give two very different proofs of this theorem.
Our first proof is carried out by showing that, for any column vector y, Pny
tends to a constant vector. As indicated in Section 11.3, this will show that Pn
converges to a matrix with constant columns or, equivalently, to a matrix with all
rows the same.
The following lemma says that if an r-by-r transition matrix has no zero entries,
and y is any column vector with r entries, then the vector Py has entries which are
“closer together” than the entries are in y.
16E. G. Coffman, J. T. Kaduta, and L. A. Shepp, “On the Asymptotic Optimality of First-
Storage Allocation,” IEEE Trans. Software Engineering, vol. II (1985), pp. 235-239.
448 CHAPTER 11. MARKOV CHAINS
Lemma 11.1 Let P be an r-by-r transition matrix with no zero entries. Let d be
the smallest entry of the matrix. Let y be a column vector with r components, the
largest of which is M0 and the smallest m0. Let M1 and m1 be the largest and
smallest component, respectively, of the vector Py. Then
M1 − m1 ≤ (1 − 2d)(M0 − m0) .
Proof. In the discussion following Theorem11.7, it was noted that each entry in the
vector Py is a weighted average of the entries in y. The largest weighted average
that could be obtained in the present case would occur if all but one of the entries
of y have value M0 and one entry has value m0, and this one small entry is weighted
by the smallest possible weight, namely d. In this case, the weighted average would
equal
dm0 + (1 − d)M0 .
Similarly, the smallest possible weighted average equals
dM0 + (1 − d)m0 .
Thus,
M1 − m1 ≤ dm0 + (1 − d)M0 − dM0 + (1 − d)m0
= (1 − 2d)(M0 − m0) .
This completes the proof of the lemma. 2
We turn now to the proof of the fundamental limit theorem for regular Markov
chains.
Theorem 11.13 (Fundamental Limit Theorem for Regular Chains) If P is
the transition matrix for a regular Markov chain, then
lim Pn = W ,
n→∞
where W is matrix with all rows equal. Furthermore, all entries in W are strictly
positive.
Proof. We prove this theorem for the special case that P has no 0 entries. The
extension to the general case is indicated in Exercise 5. Let y be any r-component
column vector, where r is the number of states of the chain. We assume that
r > 1, since otherwise the theorem is trivial. Let Mn and mn be, respectively,
the maximum and minimum components of the vector Pn y. The vector Pny is
obtained from the vector Pn−1y by multiplying on the left by the matrix P. Hence
each component of Pny is an average of the components of Pn−1y. Thus
M0 ≥ M1 ≥ M2 ≥ · · ·
11.4. FUNDAMENTAL LIMIT THEOREM 449
and
m0 ≤ m1 ≤ m2 ≤ · · · .
Each sequence is monotone and bounded:
m0 ≤ mn ≤ Mn ≤ M0 .
Hence, each of these sequences will have a limit as n tends to infinity.
Let M be the limit of Mn and m the limit of mn. We know that m ≤ M . We
shall prove that M − m = 0. This will be the case if Mn − mn tends to 0. Let d
be the smallest element of P. Since all entries of P are strictly positive, we have
d > 0. By our lemma
Mn − mn ≤ (1 − 2d)(Mn−1 − mn−1) .
From this we see that
Mn − mn ≤ (1 − 2d)n(M0 − m0) .
Since r ≥ 2, we must have d ≤ 1/2, so 0 ≤ 1 − 2d < 1, so the difference Mn − mn
tends to 0 as n tends to infinity. Since every component of Pny lies between
mn and Mn, each component must approach the same number u = M = m. This
shows that
lim Pny = u ,
n→∞
where u is a column vector all of whose components equal u.
Now let y be the vector with jth component equal to 1 and all other components
equal to 0. Then Pny is the jth column of Pn. Doing this for each j proves that the
columns of Pn approach constant column vectors. That is, the rows of Pn approach
a common row vector w, or,
lim Pn = W .
n→∞
It remains to show that all entries in W are strictly positive. As before, let y
be the vector with jth component equal to 1 and all other components equal to 0.
Then Py is the jth column of P, and this column has all entries strictly positive.
The minimum component of the vector Py was defined to be m1, hence m1 > 0.
Since m1 ≤ m, we have m > 0. Note finally that this value of m is just the jth
component of w, so all components of w are strictly positive. 2
Doeblin’s Proof
We give now a very different proof of the main part of the fundamental limit theorem
for regular Markov chains. This proof was first given by Doeblin,17 a brilliant young
mathematician who was killed in his twenties in the Second World War.
17W. Doeblin, “Expos´e de la Th´eorie des Chaines Simple Constantes de Markov `a un Nombre
Fini d’Etats,” Rev. Mach. de l’Union Interbalkanique, vol. 2 (1937), pp. 77–105.
450 CHAPTER 11. MARKOV CHAINS
Theorem 11.14 Let P be the transition matrix for a regular Markov chain with
fixed vector w. Then for any initial probability vector u, uPn → w as n → ∞.
Proof. Let X0, X1, . . . be a Markov chain with transition matrix P started in
state si. Let Y0, Y1, . . . be a Markov chain with transition probability P started
with initial probabilities given by w. The X and Y processes are run independently
of each other.
We consider also a third Markov chain P∗ which consists of watching both the
X and Y processes. The states for P∗ are pairs (si, sj). The transition probabilities
are given by
P∗[(i, j), (k, l)] = P(i, k) · P(j, l) .
Since P is regular there is an N such that PN (i, j) > 0 for all i and j. Thus for the
P∗ chain it is also possible to go from any state (si, sj) to any other state (sk, sl)
in at most N steps. That is P∗ is also a regular Markov chain.
We know that a regular Markov chain will reach any state in a finite time. Let T
be the first time the the chain P∗ is in a state of the form (sk, sk). In other words,
T is the first time that the X and the Y processes are in the same state. Then we
have shown that
P [T > n] → 0 as n → ∞ .
If we watch the X and Y processes after the first time they are in the same state
we would not predict any difference in their long range behavior. Since this will
happen no matter how we started these two processes, it seems clear that the long
range behaviour should not depend upon the starting state. We now show that this
is true.
We first note that if n ≥ T , then since X and Y are both in the same state at
time T ,
P (Xn = j | n ≥ T ) = P (Yn = j | n ≥ T ) .
If we multiply both sides of this equation by P (n ≥ T ), we obtain
P (Xn = j, n ≥ T ) = P (Yn = j, n ≥ T ) . (11.1)
We know that for all n,
P (Yn = j) = wj .
But
P (Yn = j) = P (Yn = j, n ≥ T ) + P (Yn = j, n < T ) ,
and the second summand on the right-hand side of this equation goes to 0 as n goes
to ∞, since P (n < T ) goes to 0 as n goes to ∞. So,
P (Yn = j, n ≥ T ) → wj ,
as n goes to ∞. From Equation 11.1, we see that
P (Xn = j, n ≥ T ) → wj ,
11.4. FUNDAMENTAL LIMIT THEOREM 451
as n goes to ∞. But by similar reasoning to that used above, the difference between
this last expression and P (Xn = j) goes to 0 as n goes to ∞. Therefore,
P (Xn = j) → wj ,
as n goes to ∞. This completes the proof. 2
In the above proof, we have said nothing about the rate at which the distributions
of the Xn’s approach the fixed distribution w. In fact, it can be shown that18
r
| P (Xn = j) − wj |≤ 2P (T > n) .
j=1
The left-hand side of this inequality can be viewed as the distance between the
distribution of the Markov chain after n steps, starting in state si, and the limiting
distribution w.
Exercises
1 Define P and y by
P= .5 .5 , y= 1 .
.25 .75 0
Compute Py, P2y, and P4y and show that the results are approaching a
constant vector. What is this vector?
2 Let P be a regular r × r transition matrix and y any r-component column
vector. Show that the value of the limiting constant vector for Pny is wy.
3 Let 1 0 0
P = .25 0 .75
001
be a transition matrix of a Markov chain. Find two fixed vectors of P that are
linearly independent. Does this show that the Markov chain is not regular?
4 Describe the set of all fixed column vectors for the chain given in Exercise 3.
5 The theorem that Pn → W was proved only for the case that P has no zero
entries. Fill in the details of the following extension to the case that P is
regular. Since P is regular, for some N, PN has no zeros. Thus, the proof
given shows that MnN − mnN approaches 0 as n tends to infinity. However,
the difference Mn − mn can never increase. (Why?) Hence, if we know that
the differences obtained by looking at every N th time tend to 0, then the
entire sequence must also tend to 0.
6 Let P be a regular transition matrix and let w be the unique non-zero fixed
vector of P. Show that no entry of w is 0.
18T. Lindvall, Lectures on the Coupling Method (New York: Wiley 1992).
452 CHAPTER 11. MARKOV CHAINS
7 Here is a trick to try on your friends. Shuffle a deck of cards and deal them
out one at a time. Count the face cards each as ten. Ask your friend to look
at one of the first ten cards; if this card is a six, she is to look at the card that
turns up six cards later; if this card is a three, she is to look at the card that
turns up three cards later, and so forth. Eventually she will reach a point
where she is to look at a card that turns up x cards later but there are not
x cards left. You then tell her the last card that she looked at even though
you did not know her starting point. You tell her you do this by watching
her, and she cannot disguise the times that she looks at the cards. In fact you
just do the same procedure and, even though you do not start at the same
point as she does, you will most likely end at the same point. Why?
8 Write a program to play the game in Exercise 7.
9 (Suggested by Peter Doyle) In the proof of Theorem 11.14, we assumed the
existence of a fixed vector w. To avoid this assumption, beef up the coupling
argument to show (without assuming the existence of a stationary distribution
w) that for appropriate constants C and r < 1, the distance between αP n
and βP n is at most Crn for any starting distributions α and β. Apply this
in the case where β = αP to conclude that the sequence αP n is a Cauchy
sequence, and that its limit is a matrix W whose rows are all equal to a
probability vector w with wP = w. Note that the distance between αP n and
w is at most Crn, so in freeing ourselves from the assumption about having
a fixed vector we’ve proved that the convergence to equilibrium takes place
exponentially fast.
11.5 Mean First Passage Time for Ergodic Chains
In this section we consider two closely related descriptive quantities of interest for
ergodic chains: the mean time to return to a state and the mean time to go from
one state to another state.
Let P be the transition matrix of an ergodic chain with states s1, s2, . . . , sr. Let
w = (w1, w2, . . . , wr) be the unique probability vector such that wP = w. Then,
by the Law of Large Numbers for Markov chains, in the long run the process will
spend a fraction wj of the time in state sj. Thus, if we start in any state, the chain
will eventually reach state sj; in fact, it will be in state sj infinitely often.
Another way to see this is the following: Form a new Markov chain by making
sj an absorbing state, that is, define pjj = 1. If we start at any state other than sj,
this new process will behave exactly like the original chain up to the first time that
state sj is reached. Since the original chain was an ergodic chain, it was possible
to reach sj from any other state. Thus the new chain is an absorbing chain with a
single absorbing state sj that will eventually be reached. So if we start the original
chain at a state si with i = j, we will eventually reach the state sj.
Let N be the fundamental matrix for the new chain. The entries of N give the
expected number of times in each state before absorption. In terms of the original
11.5. MEAN FIRST PASSAGE TIME 453
12
3
65 4
78 9
Figure 11.5: The maze problem.
chain, these quantities give the expected number of times in each of the states before
reaching state sj for the first time. The ith component of the vector Nc gives the
expected number of steps before absorption in the new chain, starting in state si.
In terms of the old chain, this is the expected number of steps required to reach
state sj for the first time starting at state si.
Mean First Passage Time
Definition 11.7 If an ergodic Markov chain is started in state si, the expected
number of steps to reach state sj for the first time is called the mean first passage
time from si to sj. It is denoted by mij. By convention mii = 0. 2
Example 11.24 Let us return to the maze example (Example 11.22). We shall
make this ergodic chain into an absorbing chain by making state 5 an absorbing
state. For example, we might assume that food is placed in the center of the maze
and once the rat finds the food, he stays to enjoy it (see Figure 11.5).
The new transition matrix in canonical form is
123467895
1 0 1/2 0 0 1/2 0 0 0 0
2 1/3 0 1/3 0 0 0 0 0 1/3
3 0 1/2 0 1/2 0 0 0 0 0
4 0 0 1/3 0 0 1/3 0 1/3 1/3
P = 6 1/3 0 0 0 0 0 0 0 1/3 .
7 0 0 0 0 1/2 0 1/2 0 0
8 0 0 0 0 0 1/3 0 1/3 1/3
9 0 0 0 1/2 0 0 1/2 0 0
50 0 0 0 0 0 0 0 1
454 CHAPTER 11. MARKOV CHAINS
If we compute the fundamental matrix N, we obtain
14 9 4 3 9 4 3 2
6 14 6 4 4 2 2 2
9 14 9 3 2 3
4 6 14 2 2 4
4 4 2 2 14 6 4 4
3 2 3 9 14 9
2 2 4 4 6 14
N = 1 2 6 .
8 6 2
4 4
2 6
2 3 4 9 3 4 9 14
The expected time to absorption for different starting states is given by the vec-
tor Nc, where
6
5
6
Nc = 5 .
5
6
5
6
We see that, starting from compartment 1, it will take on the average six steps
to reach food. It is clear from symmetry that we should get the same answer for
starting at state 3, 7, or 9. It is also clear that it should take one more step,
starting at one of these states, than it would starting at 2, 4, 6, or 8. Some of the
results obtained from N are not so obvious. For instance, we note that the expected
number of times in the starting state is 14/8 regardless of the state in which we
start. 2
Mean Recurrence Time
A quantity that is closely related to the mean first passage time is the mean recur-
rence time, defined as follows. Assume that we start in state si; consider the length
of time before we return to si for the first time. It is clear that we must return,
since we either stay at si the first step or go to some other state sj, and from any
other state sj, we will eventually reach si because the chain is ergodic.
Definition 11.8 If an ergodic Markov chain is started in state si, the expected
number of steps to return to si for the first time is the mean recurrence time for si.
It is denoted by ri. 2
We need to develop some basic properties of the mean first passage time. Con-
sider the mean first passage time from si to sj; assume that i = j. This may be
computed as follows: take the expected number of steps required given the outcome
of the first step, multiply by the probability that this outcome occurs, and add. If
the first step is to sj, the expected number of steps required is 1; if it is to some
11.5. MEAN FIRST PASSAGE TIME 455
other state sk, the expected number of steps required is mkj plus 1 for the step
already taken. Thus,
mij = pij + pik(mkj + 1) ,
k=j
or, since k pik = 1,
mij = 1 + pikmkj . (11.2)
k=j
Similarly, starting in si, it must take at least one step to return. Considering
all possible first steps gives us
ri = pik(mki + 1) (11.3)
(11.4)
k
= 1 + pikmki .
k
Mean First Passage Matrix and Mean Recurrence Matrix
Let us now define two matrices M and D. The ijth entry mij of M is the mean first
passage time to go from si to sj if i = j; the diagonal entries are 0. The matrix M
is called the mean first passage matrix. The matrix D is the matrix with all entries
0 except the diagonal entries dii = ri. The matrix D is called the mean recurrence
matrix. Let C be an r × r matrix with all entries 1. Using Equation 11.2 for the
case i = j and Equation 11.4 for the case i = j, we obtain the matrix equation
M = PM + C − D , (11.5)
or
(I − P)M = C − D . (11.6)
Equation 11.6 with mii = 0 implies Equations 11.2 and 11.4. We are now in a
position to prove our first basic theorem.
Theorem 11.15 For an ergodic Markov chain, the mean recurrence time for state
si is ri = 1/wi, where wi is the ith component of the fixed probability vector for
the transition matrix.
Proof. Multiplying both sides of Equation 11.6 by w and using the fact that
w(I − P) = 0
gives
wC − wD = 0 .
Here wC is a row vector with all entries 1 and wD is a row vector with ith entry
wiri. Thus
(1, 1, . . . , 1) = (w1r1, w2r2, . . . , wnrn)
and
ri = 1/wi ,
as was to be proved. 2
456 CHAPTER 11. MARKOV CHAINS
Corollary 11.1 For an ergodic Markov chain, the components of the fixed proba-
bility vector w are strictly positive.
Proof. We know that the values of ri are finite and so wi = 1/ri cannot be 0. 2
Example 11.25 In Example 11.22 we found the fixed probability vector for the
maze example to be
w = ( 1 1 1 1 1 1 1 1 1 ) .
12 8 12 8 6 8 12 8 12
Hence, the mean recurrence times are given by the reciprocals of these probabilities.
That is,
r = ( 12 8 12 8 6 8 12 8 12 ) .
2
Returning to the Land of Oz, we found that the weather in the Land of Oz could
be represented by a Markov chain with states rain, nice, and snow. In Section 11.3
we found that the limiting vector was w = (2/5, 1/5, 2/5). From this we see that
the mean number of days between rainy days is 5/2, between nice days is 5, and
between snowy days is 5/2.
Fundamental Matrix
We shall now develop a fundamental matrix for ergodic chains that will play a role
similar to that of the fundamental matrix N = (I − Q)−1 for absorbing chains. As
was the case with absorbing chains, the fundamental matrix can be used to find
a number of interesting quantities involving ergodic chains. Using this matrix, we
will give a method for calculating the mean first passage times for ergodic chains
that is easier to use than the method given above. In addition, we will state (but
not prove) the Central Limit Theorem for Markov Chains, the statement of which
uses the fundamental matrix.
We begin by considering the case that P is the transition matrix of a regular
Markov chain. Since there are no absorbing states, we might be tempted to try
Z = (I − P)−1 for a fundamental matrix. But I − P does not have an inverse. To
see this, recall that a matrix R has an inverse if and only if Rx = 0 implies x = 0.
But since Pc = c we have (I − P)c = 0, and so I − P does not have an inverse.
We recall that if we have an absorbing Markov chain, and Q is the restriction
of the transition matrix to the set of transient states, then the fundamental matrix
N could be written as
N = I + Q + Q2 + · · · .
The reason that this power series converges is that Qn → 0, so this series acts like
a convergent geometric series.
This idea might prompt one to try to find a similar series for regular chains.
Since we know that Pn → W, we might consider the series
I + (P − W) + (P2 − W) + · · · . (11.7)
11.5. MEAN FIRST PASSAGE TIME 457
We now use special properties of P and W to rewrite this series. The special
properties are: 1) PW = W, and 2) Wk = W for all positive integers k. These
facts are easy to verify, and are left as an exercise (see Exercise 22). Using these
facts, we see that
(P − W)n = n n Pn−iWi
(−1)i i
i=0
= n n Wi
i
Pn + (−1)i
i=1
= n n
W
Pn + (−1)i i
i=1
= Pn + n n W.
(−1)i i
i=1
If we expand the expression (1 − 1)n, using the Binomial Theorem, we obtain the
expression in parenthesis above, except that we have an extra term (which equals
1). Since (1 − 1)n = 0, we see that the above expression equals -1. So we have
(P − W)n = Pn − W ,
for all n ≥ 1.
We can now rewrite the series in 11.7 as
I + (P − W) + (P − W)2 + · · · .
Since the nth term in this series is equal to Pn − W, the nth term goes to 0 as n
goes to infinity. This is sufficient to show that this series converges, and sums to
the inverse of the matrix I − P + W. We call this inverse the fundamental matrix
associated with the chain, and we denote it by Z.
In the case that the chain is ergodic, but not regular, it is not true that Pn → W
as n → ∞. Nevertheless, the matrix I − P + W still has an inverse, as we will now
show.
Proposition 11.1 Let P be the transition matrix of an ergodic chain, and let W
be the matrix all of whose rows are the fixed probability row vector for P. Then
the matrix
I−P+W
has an inverse.
Proof. Let x be a column vector such that
(I − P + W)x = 0 .
To prove the proposition, it is sufficient to show that x must be the zero vector.
Multiplying this equation by w and using the fact that w(I− P) = 0 and wW = w,
we have
w(I − P + W)x = wx = 0 .
458 CHAPTER 11. MARKOV CHAINS
Therefore,
(I − P)x = 0 .
But this means that x = Px is a fixed column vector for P. By Theorem 11.10,
this can only happen if x is a constant vector. Since wx = 0, and w has strictly
positive entries, we see that x = 0. This completes the proof. 2
As in the regular case, we will call the inverse of the matrix I − P + W the
fundamental matrix for the ergodic chain with transition matrix P, and we will use
Z to denote this fundamental matrix.
Example 11.26 Let P be the transition matrix for the weather in the Land of Oz.
Then
1 0 0 1/2 1/4 1/4 2/5 1/5 2/5
I − P + W = 0 1 0 − 1/2 0 1/2 + 2/5 1/5 2/5
001 1/4 1/4 1/2 2/5 1/5 2/5
9/10 −1/20 3/20
= −1/10 6/5 −1/10 ,
3/20 −1/20 9/10
so
86/75 1/25 −14/75
Z = (I − P + W)−1 = 2/25 21/25 2/25 .
−14/75 1/25 86/75
2
Using the Fundamental Matrix to Calculate the Mean First
Passage Matrix
We shall show how one can obtain the mean first passage matrix M from the
fundamental matrix Z for an ergodic Markov chain. Before stating the theorem
which gives the first passage times, we need a few facts about Z.
Lemma 11.2 Let Z = (I − P + W)−1, and let c be a column vector of all 1’s.
Then
Zc = c ,
wZ = w ,
and
Z(I − P) = I − W .
Proof. Since Pc = c and Wc = c,
c = (I − P + W)c .
If we multiply both sides of this equation on the left by Z, we obtain
Zc = c .
11.5. MEAN FIRST PASSAGE TIME 459
Similarly, since wP = w and wW = w,
w = w(I − P + W) .
If we multiply both sides of this equation on the right by Z, we obtain
wZ = w .
Finally, we have
(I − P + W)(I − W) = I − W − P + W + W − W
= I−P .
Multiplying on the left by Z, we obtain
I − W = Z(I − P) .
This completes the proof. 2
The following theorem shows how one can obtain the mean first passage times
from the fundamental matrix.
Theorem 11.16 The mean first passage matrix M for an ergodic chain is deter-
mined from the fundamental matrix Z and the fixed row probability vector w by
mij = zjj − zij .
wj
Proof. We showed in Equation 11.6 that
(I − P)M = C − D .
Thus,
Z(I − P)M = ZC − ZD ,
and from Lemma 11.2,
Z(I − P)M = C − ZD .
Again using Lemma 11.2, we have
M − WM = C − ZD
or
M = C − ZD + WM .
From this equation, we see that
mij = 1 − zij rj + (wM)j . (11.8)
0 = 1 − zjjrj + (wM)j ,
But mjj = 0, and so
460 CHAPTER 11. MARKOV CHAINS
or (11.9)
(wM)j = zjjrj − 1 . 2
From Equations 11.8 and 11.9, we have
mij = (zjj − zij ) · rj .
Since rj = 1/wj, zjj − zij
wj
mij = .
Example 11.27 (Example 11.26 continued) In the Land of Oz example, we find
that
86/75 1/25 −14/75
Z = (I − P + W)−1 = 2/25 21/25 2/25 .
−14/75 1/25 86/75
We have also seen that w = (2/5, 1/5, 2/5). So, for example,
m12 = z22 − z12
w2
21/25 − 1/25
=
1/5
= 4,
by Theorem 11.16. Carrying out the calculations for the other entries of M, we
obtain 0 4 10/3
M = 8/3 0 8/3 .
10/3 4 0
2
Computation
The program ErgodicChain calculates the fundamental matrix, the fixed vector,
the mean recurrence matrix D, and the mean first passage matrix M. We have run
the program for the Ehrenfest urn model (Example 11.8). We obtain:
0 1 2 3 4
0 .0000 1.0000 .0000 .0000 .0000
.0000 .7500 .0000
1 .2500 .5000 .0000 .5000 .0000
.0000 .7500 .0000
P= 2 .0000 .0000 .0000 1.0000 .0000 ;
3 .0000 .2500
4 .0000 .0000
01234
w = .0625 .2500 .3750 .2500 .0625 ;
11.5. MEAN FIRST PASSAGE TIME 461
0 123 4
r = 16.0000 4.0000 2.6667 4.0000 16.0000 ;
0 1 2 3 4
0 .0000 1.0000 2.6667 6.3333 21.3333
.0000 1.6667 5.3333
1 15.0000 3.6667 .0000 3.6667 20.3333
5.3333 1.6667 .0000
M= 2 18.6667 6.3333 2.6667 1.0000 18.6667 .
3 20.3333 15.0000
4 21.3333 .0000
From the mean first passage matrix, we see that the mean time to go from 0 balls
in urn 1 to 2 balls in urn 1 is 2.6667 steps while the mean time to go from 2 balls in
urn 1 to 0 balls in urn 1 is 18.6667. This reflects the fact that the model exhibits a
central tendency. Of course, the physicist is interested in the case of a large number
of molecules, or balls, and so we should consider this example for n so large that
we cannot compute it even with a computer.
Ehrenfest Model
Example 11.28 (Example 11.23 continued) Let us consider the Ehrenfest model
(see Example 11.8) for gas diffusion for the general case of 2n balls. Every second,
one of the 2n balls is chosen at random and moved from the urn it was in to the
other urn. If there are i balls in the first urn, then with probability i/2n we take
one of them out and put it in the second urn, and with probability (2n − i)/2n we
take a ball from the second urn and put it in the first urn. At each second we let
the number i of balls in the first urn be the state of the system. Then from state i
we can pass only to state i − 1 and i + 1, and the transition probabilities are given
by
i , if j = i − 1,
2n
pij = 1− i , if j = i + 1,
2n
0 , otherwise.
This defines the transition matrix of an ergodic, non-regular Markov chain (see
Exercise 15). Here the physicist is interested in long-term predictions about the
state occupied. In Example 11.23, we gave an intuitive reason for expecting that
the fixed vector w is the binomial distribution with parameters 2n and 1/2. It is
easy to check that this is correct. So,
2n
wi = i .
22n
Thus the mean recurrence time for state i is
22n
ri = 2n .
i
462 Time forward CHAPTER 11. MARKOV CHAINS
200 400
65 600 800 1000
60
55
50
45
40
0
Time reversed
65
60
55
50
45
40
0 200 400 600 800 1000
Figure 11.6: Ehrenfest simulation.
Consider in pa√rticular the central term i = n. We ha√ve seen that this term is
approximately 1/ πn. Thus we may approximate rn by πn.
This model was used to explain the concept of reversibility in physical systems.
Assume that we let our system run until it is in equilibrium. At this point, a movie
is made, showing the system’s progress. The movie is then shown to you, and you
are asked to tell if the movie was shown in the forward or the reverse direction.
It would seem that there should always be a tendency to move toward an equal
proportion of balls so that the correct order of time should be the one with the
most transitions from i to i − 1 if i > n and i to i + 1 if i < n.
In Figure 11.6 we show the results of simulating the Ehrenfest urn model for
the case of n = 50 and 1000 time units, using the program EhrenfestUrn. The
top graph shows these results graphed in the order in which they occurred and the
bottom graph shows the same results but with time reversed. There is no apparent
difference.
11.5. MEAN FIRST PASSAGE TIME 463
We note that if we had not started in equilibrium, the two graphs would typically
look quite different. 2
Reversibility
If the Ehrenfest model is started in equilibrium, then the process has no apparent
time direction. The reason for this is that this process has a property called re-
versibility. Define Xn to be the number of balls in the left urn at step n. We can
calculate, for a general ergodic chain, the reverse transition probability:
P (Xn−1 = j|Xn = i) = P (Xn−1 = j, Xn = i)
P (Xn = i)
= P (Xn−1 = j)P (Xn = i|Xn−1 = j)
P (Xn = i)
= P (Xn−1 = j)pji .
P (Xn = i)
In general, this will depend upon n, since P (Xn = j) and also P (Xn−1 = j)
change with n. However, if we start with the vector w or wait until equilibrium is
reached, this will not be the case. Then we can define
pi∗j = wj pji
wi
as a transition matrix for the process watched with time reversed.
Let us calculate a typical transition probability for the reverse chain P∗ = {pi∗j}
in the Ehrenfest model. For example,
pi∗,i−1 = wi−1pi−1,i = 2n 2n − i + 1 22n
wi i−1 × 2n × 2n
22n i
= (2n)! × (2n − i + 1)i! (2n − i)!
(i − 1)! (2n − i + 1)! 2n(2n)!
i
= 2n = pi,i−1 .
Similar calculations for the other transition probabilities show that P∗ = P.
When this occurs the process is called reversible. Clearly, an ergodic chain is re-
versible if, and only if, for every pair of states si and sj, wipij = wjpji. In particular,
for the Ehrenfest model this means that wipi,i−1 = wi−1pi−1,i. Thus, in equilib-
rium, the pairs (i, i − 1) and (i − 1, i) should occur with the same frequency. While
many of the Markov chains that occur in applications are reversible, this is a very
strong condition. In Exercise 12 you are asked to find an example of a Markov chain
which is not reversible.
The Central Limit Theorem for Markov Chains
Suppose that we have an ergodic Markov chain with states s1, s2, . . . , sk. It is
natural to consider the distribution of the random variables Sj(n), which denotes
464 CHAPTER 11. MARKOV CHAINS
the number of times that the chain is in state sj in the first n steps. The jth
component wj of the fixed probability row vector w is the proportion of times that
the chain is in state sj in the long run. Hence, it is reasonable to conjecture that
the expected value of the random variable Sj(n), as n → ∞, is asymptotic to nwj,
and it is easy to show that this is the case (see Exercise 23).
It is also natural to ask whether there is a limiting distribution of the random
variables Sj(n). The answer is yes, and in fact, this limiting distribution is the normal
distribution. As in the case of independent trials, one must normalize these random
variables. Thus, we must subtract from Sj(n) its expected value, and then divide by
its standard deviation. In both cases, we will use the asymptotic values of these
quantities, rather than the values themselves. Thus, in the first case, we will use
the value nwj. It is not so clear what we should use in the second case. It turns
out that the quantity
σj2 = 2wj zjj − wj − wj2 (11.10)
represents the asymptotic variance. Armed with these ideas, we can state the
following theorem.
Theorem 11.17 (Central Limit Theorem for Markov Chains) For an er-
godic chain, for any real numbers r < s, we have
P r < Sj(n) − nwj < s → √1 s
nσj2 2π r
e−x2/2 dx ,
as n → ∞, for any choice of starting state, where σj2 is the quantity defined in
Equation 11.10. 2
Historical Remarks
Markov chains were introduced by Andre˘i Andreevich Markov (1856–1922) and
were named in his honor. He was a talented undergraduate who received a gold
medal for his undergraduate thesis at St. Petersburg University. Besides being
an active research mathematician and teacher, he was also active in politics and
patricipated in the liberal movement in Russia at the beginning of the twentieth
century. In 1913, when the government celebrated the 300th anniversary of the
House of Romanov family, Markov organized a counter-celebration of the 200th
anniversary of Bernoulli’s discovery of the Law of Large Numbers.
Markov was led to develop Markov chains as a natural extension of sequences
of independent random variables. In his first paper, in 1906, he proved that for a
Markov chain with positive transition probabilities and numerical states the average
of the outcomes converges to the expected value of the limiting distribution (the
fixed vector). In a later paper he proved the central limit theorem for such chains.
Writing about Markov, A. P. Youschkevitch remarks:
Markov arrived at his chains starting from the internal needs of prob-
ability theory, and he never wrote about their applications to physical
11.5. MEAN FIRST PASSAGE TIME 465
science. For him the only real examples of the chains were literary texts,
where the two states denoted the vowels and consonants.19
In a paper written in 1913,20 Markov chose a sequence of 20,000 letters from
Pushkin’s Eugene Onegin to see if this sequence can be approximately considered
a simple chain. He obtained the Markov chain with transition matrix
vowel vowel consonant .
consonant .128 .872
.663 .337
The fixed vector for this chain is (.432, .568), indicating that we should expect
about 43.2 percent vowels and 56.8 percent consonants in the novel, which was
borne out by the actual count.
Claude Shannon considered an interesting extension of this idea in his book The
Mathematical Theory of Communication,21 in which he developed the information-
theoretic concept of entropy. Shannon considers a series of Markov chain approxi-
mations to English prose. He does this first by chains in which the states are letters
and then by chains in which the states are words. For example, for the case of
words he presents first a simulation where the words are chosen independently but
with appropriate frequencies.
REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME
CAN DIFFERENT NATURAL HERE HE THE A IN CAME THE TO
OF TO EXPERT GRAY COME TO FURNISHES THE LINE MES-
SAGE HAD BE THESE.
He then notes the increased resemblence to ordinary English text when the words
are chosen as a Markov chain, in which case he obtains
THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRI-
TER THAT THE CHARACTER OF THIS POINT IS THEREFORE
ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF
WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED.
A simulation like the last one is carried out by opening a book and choosing the
first word, say it is the. Then the book is read until the word the appears again
and the word after this is chosen as the second word, which turned out to be head.
The book is then read until the word head appears again and the next word, and,
is chosen, and so on.
Other early examples of the use of Markov chains occurred in Galton’s study of
the problem of survival of family names in 1889 and in the Markov chain introduced
19See Dictionary of Scientific Biography, ed. C. C. Gillespie (New York: Scribner’s Sons, 1970),
pp. 124–130.
20A. A. Markov, “An Example of Statistical Analysis of the Text of Eugene Onegin Illustrat-
ing the Association of Trials into a Chain,” Bulletin de l’Acadamie Imperiale des Sciences de
St. Petersburg, ser. 6, vol. 7 (1913), pp. 153–162.
21C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (Urbana: Univ.
of Illinois Press, 1964).
466 CHAPTER 11. MARKOV CHAINS
by P. and T. Ehrenfest in 1907 for diffusion. Poincar´e in 1912 dicussed card shuffling
in terms of an ergodic Markov chain defined on a permutation group. Brownian
motion, a continuous time version of random walk, was introducted in 1900–1901
by L. Bachelier in his study of the stock market, and in 1905–1907 in the works of
A. Einstein and M. Smoluchowsky in their study of physical processes.
One of the first systematic studies of finite Markov chains was carried out by
M. Frechet.22 The treatment of Markov chains in terms of the two fundamental
matrices that we have used was developed by Kemeny and Snell 23 to avoid the use of
eigenvalues that one of these authors found too complex. The fundamental matrix N
occurred also in the work of J. L. Doob and others in studying the connection
between Markov processes and classical potential theory. The fundamental matrix Z
for ergodic chains appeared first in the work of Frechet, who used it to find the
limiting variance for the central limit theorem for Markov chains.
Exercises
1 Consider the Markov chain with transition matrix
1/2 1/2
P= .
1/4 3/4
Find the fundamental matrix Z for this chain. Compute the mean first passage
matrix using Z.
2 A study of the strengths of Ivy League football teams shows that if a school
has a strong team one year it is equally likely to have a strong team or average
team next year; if it has an average team, half the time it is average next year,
and if it changes it is just as likely to become strong as weak; if it is weak it
has 2/3 probability of remaining so and 1/3 of becoming average.
(a) A school has a strong team. On the average, how long will it be before
it has another strong team?
(b) A school has a weak team; how long (on the average) must the alumni
wait for a strong team?
3 Consider Example 11.4 with a = .5 and b = .75. Assume that the President
says that he or she will run. Find the expected length of time before the first
time the answer is passed on incorrectly.
4 Find the mean recurrence time for each state of Example 11.4 for a = .5 and
b = .75. Do the same for general a and b.
5 A die is rolled repeatedly. Show by the results of this section that the mean
time between occurrences of a given number is 6.
22M. Frechet, “Th´eorie des ´ev´enements en chaine dans le cas d’un nombre fini d’´etats possible,”
in Recherches th´eoriques Modernes sur le calcul des probabilit´es, vol. 2 (Paris, 1938).
23J. G. Kemeny and J. L. Snell, Finite Markov Chains.
11.5. MEAN FIRST PASSAGE TIME 467
1
2 34
56
Figure 11.7: Maze for Exercise 7.
6 For the Land of Oz example (Example 11.1), make rain into an absorbing
state and find the fundamental matrix N. Interpret the results obtained from
this chain in terms of the original chain.
7 A rat runs through the maze shown in Figure 11.7. At each step it leaves the
room it is in by choosing at random one of the doors out of the room.
(a) Give the transition matrix P for this Markov chain.
(b) Show that it is an ergodic chain but not a regular chain.
(c) Find the fixed vector.
(d) Find the expected number of steps before reaching Room 5 for the first
time, starting in Room 1.
8 Modify the program ErgodicChain so that you can compute the basic quan-
tities for the queueing example of Exercise 11.3.20. Interpret the mean recur-
rence time for state 0.
9 Consider a random walk on a circle of circumference n. The walker takes
one unit step clockwise with probability p and one unit counterclockwise with
probability q = 1 − p. Modify the program ErgodicChain to allow you to
input n and p and compute the basic quantities for this chain.
(a) For which values of n is this chain regular? ergodic?
(b) What is the limiting vector w?
(c) Find the mean first passage matrix for n = 5 and p = .5. Verify that
mij = d(n − d), where d is the clockwise distance from i to j.
10 Two players match pennies and have between them a total of 5 pennies. If at
any time one player has all of the pennies, to keep the game going, he gives
one back to the other player and the game will continue. Show that this game
can be formulated as an ergodic chain. Study this chain using the program
ErgodicChain.
468 CHAPTER 11. MARKOV CHAINS
11 Calculate the reverse transition matrix for the Land of Oz example (Exam-
ple 11.1). Is this chain reversible?
12 Give an example of a three-state ergodic Markov chain that is not reversible.
13 Let P be the transition matrix of an ergodic Markov chain and P∗ the reverse
transition matrix. Show that they have the same fixed probability vector w.
14 If P is a reversible Markov chain, is it necessarily true that the mean time
to go from state i to state j is equal to the mean time to go from state j to
state i? Hint: Try the Land of Oz example (Example 11.1).
15 Show that any ergodic Markov chain with a symmetric transition matrix (i.e.,
pij = pji) is reversible.
16 (Crowell24) Let P be the transition matrix of an ergodic Markov chain. Show
that
(I + P + · · · + Pn−1)(I − P + W) = I − Pn + nW ,
and from this show that
I + P + · · · + Pn−1 → W ,
n
as n → ∞.
17 An ergodic Markov chain is started in equilibrium (i.e., with initial probability
vector w). The mean time until the next occurrence of state si is m¯i =
k wkmki + wiri. Show that m¯i = zii/wi, by using the facts that wZ = w
and mki = (zii − zki)/wi.
18 A perpetual craps game goes on at Charley’s. Jones comes into Charley’s on
an evening when there have already been 100 plays. He plans to play until the
next time that snake eyes (a pair of ones) are rolled. Jones wonders how many
times he will play. On the one hand he realizes that the average time between
snake eyes is 36 so he should play about 18 times as he is equally likely to
have come in on either side of the halfway point between occurrences of snake
eyes. On the other hand, the dice have no memory, and so it would seem
that he would have to play for 36 more times no matter what the previous
outcomes have been. Which, if either, of Jones’s arguments do you believe?
Using the result of Exercise 17, calculate the expected to reach snake eyes, in
equilibrium, and see if this resolves the apparent paradox. If you are still in
doubt, simulate the experiment to decide which argument is correct. Can you
give an intuitive argument which explains this result?
19 Show that, for an ergodic Markov chain (see Theorem 11.16),
mij wj = zjj − 1 = K .
jj
24Private communication.
11.5. MEAN FIRST PASSAGE TIME 469
- 5 20
BC
- 30 15
A GO
Figure 11.8: Simplified Monopoly.
The second expression above shows that the number K is independent of
i. The number K is called Kemeny’s constant. A prize was offered to the
first person to give an intuitively plausible reason for the above sum to be
independent of i. (See also Exercise 24.)
20 Consider a game played as follows: You are given a regular Markov chain
with transition matrix P, fixed probability vector w, and a payoff function f
which assigns to each state si an amount fi which may be positive or negative.
Assume that wf = 0. You watch this Markov chain as it evolves, and every
time you are in state si you receive an amount fi. Show that your expected
winning after n steps can be represented by a column vector g(n), with
g(n) = (I + P + P2 + · · · + Pn)f.
Show that as n → ∞, g(n) → g with g = Zf.
21 A highly simplified game of “Monopoly” is played on a board with four squares
as shown in Figure 11.8. You start at GO. You roll a die and move clockwise
around the board a number of squares equal to the number that turns up on
the die. You collect or pay an amount indicated on the square on which you
land. You then roll the die again and move around the board in the same
manner from your last position. Using the result of Exercise 20, estimate
the amount you should expect to win in the long run playing this version of
Monopoly.
22 Show that if P is the transition matrix of a regular Markov chain, and W is
the matrix each of whose rows is the fixed probability vector corresponding
to P, then PW = W, and Wk = W for all positive integers k.
23 Assume that an ergodic Markov chain has states s1, s2, . . . , sk. Let Sj(n) denote
the number of times that the chain is in state sj in the first n steps. Let w
denote the fixed probability row vector for this chain. Show that, regardless
of the starting state, the expected value of Sj(n), divided by n, tends to wj as
n → ∞. Hint: If the chain starts in state si, then the expected value of Sj(n)
is given by the expression
n
pi(jh) .
h=0
470 CHAPTER 11. MARKOV CHAINS
24 In the course of a walk with Snell along Minnehaha Avenue in Minneapolis
in the fall of 1983, Peter Doyle25 suggested the following explanation for the
constancy of Kemeny’s constant (see Exercise 19). Choose a target state
according to the fixed vector w. Start from state i and wait until the time T
that the target state occurs for the first time. Let Ki be the expected value
of T . Observe that
Ki + wi · 1/wi = PijKj + 1 ,
j
and hence
Ki = Pij Kj .
j
By the maximum principle, Ki is a constant. Should Peter have been given
the prize?
25Private communication.
Chapter 12
Random Walks
12.1 Random Walks in Euclidean Space
In the last several chapters, we have studied sums of random variables with the goal
being to describe the distribution and density functions of the sum. In this chapter,
we shall look at sums of discrete random variables from a different perspective. We
shall be concerned with properties which can be associated with the sequence of
partial sums, such as the number of sign changes of this sequence, the number of
terms in the sequence which equal 0, and the expected size of the maximum term
in the sequence.
We begin with the following definition.
Definition 12.1 Let {Xk}k∞=1 be a sequence of independent, identically distributed
discrete random variables. For each positive integer n, we let Sn denote the sum
X1 + X2 + · · · + Xn. The sequence {Sn}∞n=1 is called a random walk. If the common
range of the Xk’s is Rm, then we say that {Sn} is a random walk in Rm. 2
We view the sequence of Xk’s as being the outcomes of independent experiments.
Since the Xk’s are independent, the probability of any particular (finite) sequence
of outcomes can be obtained by multiplying the probabilities that each Xk takes
on the specified value in the sequence. Of course, these individual probabilities are
given by the common distribution of the Xk’s. We will typically be interested in
finding probabilities for events involving the related sequence of Sn’s. Such events
can be described in terms of the Xk’s, so their probabilities can be calculated using
the above idea.
There are several ways to visualize a random walk. One can imagine that a
particle is placed at the origin in Rm at time n = 0. The sum Sn represents the
position of the particle at the end of n seconds. Thus, in the time interval [n − 1, n],
the particle moves (or jumps) from position Sn−1 to Sn. The vector representing
this motion is just Sn − Sn−1, which equals Xn. This means that in a random walk,
the jumps are independent and identically distributed. If m = 1, for example, then
one can imagine a particle on the real line that starts at the origin, and at the
end of each second, jumps one unit to the right or the left, with probabilities given
471
472 CHAPTER 12. RANDOM WALKS
by the distribution of the Xk’s. If m = 2, one can visualize the process as taking
place in a city in which the streets form square city blocks. A person starts at one
corner (i.e., at an intersection of two streets) and goes in one of the four possible
directions according to the distribution of the Xk’s. If m = 3, one might imagine
being in a jungle gym, where one is free to move in any one of six directions (left,
right, forward, backward, up, and down). Once again, the probabilities of these
movements are given by the distribution of the Xk’s.
Another model of a random walk (used mostly in the case where the range is
R1) is a game, involving two people, which consists of a sequence of independent,
identically distributed moves. The sum Sn represents the score of the first person,
say, after n moves, with the assumption that the score of the second person is
−Sn. For example, two people might be flipping coins, with a match or non-match
representing +1 or −1, respectively, for the first player. Or, perhaps one coin is
being flipped, with a head or tail representing +1 or −1, respectively, for the first
player.
Random Walks on the Real Line
We shall first consider the simplest non-trivial case of a random walk in R1, namely
the case where the common distribution function of the random variables Xn is
given by
fX (x) = 1/2, if x = ±1,
0, otherwise.
This situation corresponds to a fair coin being flipped, with Sn representing the
number of heads minus the number of tails which occur in the first n flips. We note
that in this situation, all paths of length n have the same probability, namely 2−n.
It is sometimes instructive to represent a random walk as a polygonal line, or
path, in the plane, where the horizontal axis represents time and the vertical axis
represents the value of Sn. Given a sequence {Sn} of partial sums, we first plot the
points (n, Sn), and then for each k < n, we connect (k, Sk) and (k + 1, Sk+1) with a
straight line segment. The length of a path is just the difference in the time values
of the beginning and ending points on the path. The reader is referred to Figure
12.1. This figure, and the process it illustrates, are identical with the example,
given in Chapter 1, of two people playing heads or tails.
Returns and First Returns
We say that an equalization has occurred, or there is a return to the origin at time
n, if Sn = 0. We note that this can only occur if n is an even integer. To calculate
the probability of an equalization at time 2m, we need only count the number of
paths of length 2m which begin and end at the origin. The number of such paths
is clearly
2m
.
m
Since each path has probability 2−2m, we have the following theorem.
12.1. RANDOM WALKS IN EUCLIDEAN SPACE 473
10
8
6
4
2
5 10 15 20 25 30 35 40
-2
-4
-6
-8
-10
Figure 12.1: A random walk of length 40.
Theorem 12.1 The probability of a return to the origin at time 2m is given by
u2m = 2m 2−2m .
m
The probability of a return to the origin at an odd time is 0. 2
A random walk is said to have a first return to the origin at time 2m if m > 0, and
S2k = 0 for all k < m. In Figure 12.1, the first return occurs at time 2. We define
f2m to be the probability of this event. (We also define f0 = 0.) One can think
of the expression f2m22m as the number of paths of length 2m between the points
(0, 0) and (2m, 0) that do not touch the horizontal axis except at the endpoints.
Using this idea, it is easy to prove the following theorem.
Theorem 12.2 For n ≥ 1, the probabilities {u2k} and {f2k} are related by the
equation
u2n = f0u2n + f2u2n−2 + · · · + f2nu0 .
Proof. There are u2n22n paths of length 2n which have endpoints (0, 0) and (2n, 0).
The collection of such paths can be partitioned into n sets, depending upon the time
of the first return to the origin. A path in this collection which has a first return to
the origin at time 2k consists of an initial segment from (0, 0) to (2k, 0), in which
no interior points are on the horizontal axis, and a terminal segment from (2k, 0)
to (2n, 0), with no further restrictions on this segment. Thus, the number of paths
in the collection which have a first return to the origin at time 2k is given by
f2k22ku2n−2k22n−2k = f2ku2n−2k22n .
If we sum over k, we obtain the equation
u2n22n = f0u2n22n + f2u2n−222n + · · · + f2nu022n .
Dividing both sides of this equation by 22n completes the proof. 2
474 CHAPTER 12. RANDOM WALKS
The expression in the right-hand side of the above theorem should remind the reader
of a sum that appeared in Definition 7.1 of the convolution of two distributions. The
convolution of two sequences is defined in a similar manner. The above theorem
says that the sequence {u2n} is the convolution of itself and the sequence {f2n}.
Thus, if we represent each of these sequences by an ordinary generating function,
then we can use the above relationship to determine the value f2n.
Theorem 12.3 For m ≥ 1, the probability of a first return to the origin at time
2m is given by
f2m = u2m = 2m .
2m − 1 m
(2m − 1)22m
Proof. We begin by defining the generating functions
∞
U (x) = u2mxm
m=0
and ∞
Theorem 12.2 says that
F (x) = f2mxm .
(12.1)
m=0
U (x) = 1 + U (x)F (x) .
(The presence of the 1 on the right-hand side is due to the fact that u0 is defined
to be 1, but Theorem 12.2 only holds for m ≥ 1.) We note that both generating
functions certainly converge on the interval (−1, 1), since all of the coefficients are at
most 1 in absolute value. Thus, we can solve the above equation for F (x), obtaining
F (x) = U (x) − 1 .
U (x)
Now, if we can find a closed-form expression for the function U (x), we will also have
a closed-form expression for F (x). From Theorem 12.1, we have
∞ 2m 2−2mxm .
m
U (x) =
m=0
In Wilf,1 we find that ∞
√1 = 2m xm .
1 − 4x m
m=0
The reader is asked to prove this statement in Exercise 1. If we replace x by x/4
in the last equation, we see that
U (x) = √ 1 .
1−x
1H. S. Wilf, Generatingfunctionology, (Boston: Academic Press, 1990), p. 50.
12.1. RANDOM WALKS IN EUCLIDEAN SPACE 475
Therefore, we have
U (x) − 1
F (x) =
U (x)
(1 − x)−1/2 − 1
= (1 − x)−1/2
= 1 − (1 − x)1/2 .
Although it is possible to compute the value of f2m using the Binomial Theorem,
it is easier to note that F (x) = U (x)/2, so that the coefficients f2m can be found
by integrating the series for U (x). We obtain, for m ≥ 1,
f2m = u2m−2
2m
2m−2
= m−1
m22m−1
2m
= m
(2m − 1)22m
= u2m ,
2m − 1
since 2m − 2
m 2m
m − 1 = 2(2m − 1) m .
This completes the proof of the theorem. 2
Probability of Eventual Return
In the symmetric random walk process in Rm, what is the probability that the
particle eventually returns to the origin? We first examine this question in the case
that m = 1, and then we consider the general case. The results in the next two
examples are due to Po´lya.2
Example 12.1 (Eventual Return in R1) One has to approach the idea of eventual
return with some care, since the sample space seems to be the set of all walks of
infinite length, and this set is non-denumerable. To avoid difficulties, we will define
wn to be the probability that a first return has occurred no later than time n. Thus,
wn concerns the sample space of all walks of length n, which is a finite set. In terms
of the wn’s, it is reasonable to define the probability that the particle eventually
returns to the origin to be
w∗ = lim wn .
n→∞
This limit clearly exists and is at most one, since the sequence {wn}∞n=1 is an
increasing sequence, and all of its terms are at most one.
2G. Po´lya, “U¨ ber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im
Strassennetz,” Math. Ann., vol. 84 (1921), pp. 149-160.
476 CHAPTER 12. RANDOM WALKS
In terms of the fn probabilities, we see that
n
w2n = f2i .
i=1
Thus, ∞
w∗ = f2i .
i=1
In the proof of Theorem 12.3, the generating function
∞
F (x) = f2mxm
m=0
was introduced. There it was noted that this series converges for x ∈ (−1, 1). In
fact, it is possible to show that this series also converges for x = ±1 by using
Exercise 4, together with the fact that
f2m = u2m .
2m − 1
(This fact was proved in the proof of Theorem 12.3.) Since we also know that
F (x) = 1 − (1 − x)1/2 ,
we see that
w∗ = F (1) = 1 .
Thus, with probability one, the particle returns to the origin.
An alternative proof of the fact that w∗ = 1 can be obtained by using the results
in Exercise 2. 2
Example 12.2 (Eventual Return in Rm) We now turn our attention to the case
that the random walk takes place in more than one dimension. We define f2(nm) to
be the probability that the first return to the origin in Rm occurs at time 2n. The
quantity u2(mn ) is defined in a similar manner. Thus, f2(n1) and u(21n) equal f2n and u2n,
which were defined earlier. If, in addition, we define u0(m) = 1 and f0(m) = 0, then
one can mimic the proof of Theorem 12.2, and show that for all m ≥ 1,
u(2mn ) = f0(m)u2(mn ) + f2(m)u2(mn−) 2 + · · · + f2(nm)u0(m) . (12.2)
We continue to generalize previous work by defining
∞
U (m)(x) = u(2mn )xn
n=0
and
∞
F (m)(x) = f2(nm)xn .
n=0
12.1. RANDOM WALKS IN EUCLIDEAN SPACE 477
Then, by using Equation 12.2, we see that
U (m)(x) = 1 + U (m)(x)F (m)(x) ,
as before. These functions will always converge in the interval (−1, 1), since all of
their coefficients are at most one in magnitude. In fact, since
∞
w∗(m) = f2(nm) ≤ 1
n=0
for all m, the series for F (m)(x) converges at x = 1 as well, and F (m)(x) is left-
continuous at x = 1, i.e.,
lim F (m)(x) = F (m)(1) .
x↑1
Thus, we have
w∗(m) = lim F (m)(x) = lim U (m)(x) − 1 , (12.3)
U (m)(x)
x↑1 x↑1
so to determine w∗(m), it suffices to determine
lim U (m)(x) .
x↑1
We let u(m) denote this limit.
We claim that ∞
u(m) = u2(mn ) .
n=0
(This claim is reasonable; it says that to find out what happens to the function
U (m)(x) at x = 1, just let x = 1 in the power series for U (m)(x).) To prove the
claim, we note that the coefficients u(2mn ) are non-negative, so U (m)(x) increases
monotonically on the interval [0, 1). Thus, for each K, we have
K∞
u(2mn ) ≤ lim U (m)(x) = u(m) ≤ u(2mn ) .
n=0 x↑1 n=0
By letting K → ∞, we see that
∞
u(m) = u2(mn ) .
2n
This establishes the claim.
From Equation 12.3, we see that if u(m) < ∞, then the probability of an eventual
return is u(m) − 1
u(m) ,
while if u(m) = ∞, then the probability of eventual return is 1.
To complete the example, we must estimate the sum
∞
u2(mn ) .
n=0
478 CHAPTER 12. RANDOM WALKS
In Exercise 12, the reader is asked to show that
u2(2n) = 1 2n 2
42n .
n
Using Stirling’s Formula, it is easy to show that (see Exercise 13)
2n ∼ √22n ,
n πn
so 1
πn
u2(2n) ∼ .
From this it follows easily that ∞
u2(2n)
n=0
diverges, so w∗(2) = 1, i.e., in R2, the probability of an eventual return is 1.
When m = 3, Exercise 12 shows that
1 2n 1 n! 2
22n n
u2(3n) = 3n j!k!(n − j − k)! .
j,k
Let M denote the largest value of
1 n!
3n j!k!(n − j − k)! ,
over all non-negative values of j and k with j + k ≤ n. It is easy, using Stirling’s
Formula, to show that M∼c ,
n
for some constant c. Thus, we have
u(23n) ≤ 1 2n M n!
22n n 3n j!k!(n − j − k)! .
j,k
Using Exercise 14, one can show that the right-hand expression is at most
c
n3/2 ,
where c is a constant. Thus, ∞
u2(3n)
n=0
converges, so w∗(3) is strictly less than one. This means that in R3, the probability of
an eventual return to the origin is strictly less than one (in fact, it is approximately
.34).
One may summarize these results by stating that one should not get drunk in
more than two dimensions. 2
12.1. RANDOM WALKS IN EUCLIDEAN SPACE 479
Expected Number of Equalizations
We now give another example of the use of generating functions to find a general
formula for terms in a sequence, where the sequence is related by recursion relations
to other sequences. Exercise 9 gives still another example.
Example 12.3 (Expected Number of Equalizations) In this example, we will de-
rive a formula for the expected number of equalizations in a random walk of length
2m. As in the proof of Theorem 12.3, the method has four main parts. First, a
recursion is found which relates the mth term in the unknown sequence to earlier
terms in the same sequence and to terms in other (known) sequences. An exam-
ple of such a recursion is given in Theorem 12.2. Second, the recursion is used
to derive a functional equation involving the generating functions of the unknown
sequence and one or more known sequences. Equation 12.1 is an example of such
a functional equation. Third, the functional equation is solved for the unknown
generating function. Last, using a device such as the Binomial Theorem, integra-
tion, or differentiation, a formula for the mth coefficient of the unknown generating
function is found.
We begin by defining g2m to be the number of equalizations among all of the
random walks of length 2m. (For each random walk, we disregard the equalization
at time 0.) We define g0 = 0. Since the number of walks of length 2m equals 22m,
the expected number of equalizations among all such random walks is g2m/22m.
Next, we define the generating function G(x):
∞
G(x) = g2kxk .
k=0
Now we need to find a recursion which relates the sequence {g2k} to one or both of
the known sequences {f2k} and {u2k}. We consider m to be a fixed positive integer,
and consider the set of all paths of length 2m as the disjoint union
E2 ∪ E4 ∪ · · · ∪ E2m ∪ H ,
where E2k is the set of all paths of length 2m with first equalization at time 2k,
and H is the set of all paths of length 2m with no equalization. It is easy to show
(see Exercise 3) that
|E2k| = f2k22m .
We claim that the number of equalizations among all paths belonging to the set
E2k is equal to
|E2k| + 22kf2kg2m−2k . (12.4)
Each path in E2k has one equalization at time 2k, so the total number of such
equalizations is just |E2k|. This is the first summand in expression Equation 12.4.
There are 22kf2k different initial segments of length 2k among the paths in E2k.
Each of these initial segments can be augmented to a path of length 2m in 22m−2k
ways, by adjoining all possible paths of length 2m−2k. The number of equalizations
obtained by adjoining all of these paths to any one initial segment is g2m−2k, by
480 CHAPTER 12. RANDOM WALKS
definition. This gives the second summand in Equation 12.4. Since k can range
from 1 to m, we obtain the recursion
m
g2m = |E2k| + 22kf2kg2m−2k . (12.5)
k=1
The second summand in the typical term above should remind the reader of a
convolution. In fact, if we multiply the generating function G(x) by the generating
function ∞
F (4x) = 22kf2kxk ,
k=0
the coefficient of xm equals
m
22kf2kg2m−2k .
k=0
Thus, the product G(x)F (4x) is part of the functional equation that we are seeking.
The first summand in the typical term in Equation 12.5 gives rise to the sum
m
22m f2k .
k=1
From Exercise 2, we see that this sum is just (1 − u2m)22m. Thus, we need to create
a generating function whose mth coefficient is this term; this generating function is
∞
(1 − u2m)22mxm ,
m=0
or
∞∞
22mxm − u2m22mxm .
m=0 m=0
The first sum is just (1 − 4x)−1, and the second sum is U (4x). So, the functional
equation which we have been seeking is
G(x) = F (4x)G(x) + 1 1 − U (4x) .
− 4x
If we solve this recursion for G(x), and simplify, we obtain
G(x) = 1 − 1 . (12.6)
(1 − 4x)3/2 (1 − 4x)
We now need to find a formula for the coefficient of xm. The first summand in
Equation 12.6 is (1/2)U (4x), so the coefficient of xm in this function is
u2m+222m+1(m + 1) .
The second summand in Equation 12.6 is the sum of a geometric series with common
ratio 4x, so the coefficient of xm is 22m. Thus, we obtain
12.1. RANDOM WALKS IN EUCLIDEAN SPACE 481
g2m = u2m+222m+1(m + 1) − 22m
= 1 2m + 2 (m + 1) − 22m .
2 m+1
We recall that the quotient g2m/22m is the expected number of equalizations
among all paths of length 2m. Using Exercise 4, it is easy to show that
g2m ∼ 2√
22m 2m .
π
In particular, this means that the average number of equalizations among all paths
of length 4m is not twice the average number of equalizations among all paths of
length 2m. In order for the average number of equalizations to double, one must
quadruple the lengths of the random walks. 2
It is interesting to note that if we define
Mn = max Sk ,
0≤k≤n
then we have
E(Mn) ∼ 2√
n.
π
This means that the expected number of equalizations and the expected maximum
value for random walks of length n are asymptotically equal as n → ∞. (In fact,
it can be shown that the two expected values differ by at most 1/2 for all positive
integers n. See Exercise 9.)
Exercises
1 Using the Binomial Theorem, show that
√1 ∞ 2m xm .
1 − 4x m
=
m=0
What is the interval of convergence of this power series?
2 (a) Show that for m ≥ 1,
f2m = u2m−2 − u2m .
(b) Using part (a), find a closed-form expression for the sum
f2 + f4 + · · · + f2m .
(c) Using part (b), show that
∞
f2m = 1 .
m=1
(One can also obtain this statement from the fact that
F (x) = 1 − (1 − x)1/2 .)
482 CHAPTER 12. RANDOM WALKS
(d) Using parts (a) and (b), show that the probability of no equalization in
the first 2m outcomes equals the probability of an equalization at time
2m.
3 Using the notation of Example 12.3, show that
|E2k| = f2k22m .
4 Using Stirling’s Formula, show that
u2m ∼ √1 .
πm
5 A lead change in a random walk occurs at time 2k if S2k−1 and S2k+1 are of
opposite sign.
(a) Give a rigorous argument which proves that among all walks of length
2m that have an equalization at time 2k, exactly half have a lead change
at time 2k.
(b) Deduce that the total number of lead changes among all walks of length
2m equals
1 − u2m) .
2 (g2m
(c) Find an asymptotic expression for the average number of lead changes
in a random walk of length 2m.
6 (a) Show that the probability that a random walk of length 2m has a last
return to the origin at time 2k, where 0 ≤ k ≤ m, equals
2k 2m−2k
k m−k = u2ku2m−2k .
22m
(The case k = 0 consists of all paths that do not return to the origin at
any positive time.) Hint: A path whose last return to the origin occurs
at time 2k consists of two paths glued together, one path of which is of
length 2k and which begins and ends at the origin, and the other path
of which is of length 2m − 2k and which begins at the origin but never
returns to the origin. Both types of paths can be counted using quantities
which appear in this section.
(b) Using part (a), show that if m is odd, the probability that a walk of
length 2m has no equalization in the last m outcomes is equal to 1/2,
regardless of the value of m. Hint: The answer to part a) is symmetric
in k and m − k.
7 Show that the probability of no equalization in a walk of length 2m equals
u2m.
12.1. RANDOM WALKS IN EUCLIDEAN SPACE 483
*8 Show that
P (S1 ≥ 0, S2 ≥ 0, . . . , S2m ≥ 0) = u2m .
Hint: First explain why
P (S1 > 0, S2 > 0, . . . , S2m > 0)
1
= 2 P (S1 = 0, S2 = 0, . . . , S2m = 0) .
Then use Exercise 7, together with the observation that if no equalization
occurs in the first 2m outcomes, then the path goes through the point (1, 1)
and remains on or above the horizontal line x = 1.
*9 In Feller,3 one finds the following theorem: Let Mn be the random variable
which gives the maximum value of Sk, for 1 ≤ k ≤ n. Define
pn,r = n 2−n .
n+r
2
If r ≥ 0, then
P (Mn = r) = pn,r , if r ≡ n (mod 2),
pn,r+1 , if r ≡ n (mod 2).
(a) Using this theorem, show that
1 m 2m ,
E(M2m) = 22m m+k
(4k − 1)
k=1
and if n = 2m + 1, then
1m 2m + 1 .
E(M2m+1) = 22m+1 (4k + 1)
m+k+1
k=0
(b) For m ≥ 1, define rm = m 2m
and k
By using the identity k=1 m+k
m 2m + 1
sm = k .
m+k+1
k=1
n n−1 n−1
k = k−1 + k ,
show that 1 2m
sm = 2rm − 2 m
22m −
3W. Feller, Introduction to Probability Theory and its Applications, vol. I, 3rd ed. (New York:
John Wiley & Sons, 1968).
484 CHAPTER 12. RANDOM WALKS
and rm = 2sm−1 + 1 22m−1 ,
if m ≥ 2. 2
(c) Define the generating functions
∞
R(x) = rkxk
k=1
and ∞
Show that
S(x) = skxk .
k=1
S(x) = 2R(x) − 1 1 1 √
2 1 − 4x + 1 − 4x
2
and
1
R(x) = 2xS(x) + x 1 − 4x .
(d) Show that x
and R(x) = (1 − 4x)3/2 ,
11 11
S(x) = − .
2 (1 − 4x)3/2 2 1 − 4x
(e) Show that 2m − 1
and rm = m m − 1 ,
sm = 1 2m + 1 − 1 (22m) .
(m + 1) 2
2 m
(f) Show that
m 2m 1 2m 1
E(M2m) = 22m−1 + 22m+1 −,
m m 2
and
m + 1 2m + 2 1
E(M2m+1) = 22m+1 −.
m+1 2
The reader should compare these formulas with the expression for
g2m/2(2m) in Example 12.3.
12.1. RANDOM WALKS IN EUCLIDEAN SPACE 485
*10 (from K. Levasseur4) A parent and his child play the following game. A deck
of 2n cards, n red and n black, is shuffled. The cards are turned up one at a
time. Before each card is turned up, the parent and the child guess whether
it will be red or black. Whoever makes more correct guesses wins the game.
The child is assumed to guess each color with the same probability, so she
will have a score of n, on average. The parent keeps track of how many cards
of each color have already been turned up. If more black cards, say, than
red cards remain in the deck, then the parent will guess black, while if an
equal number of each color remain, then the parent guesses each color with
probability 1/2. What is the expected number of correct guesses that will be
made by the parent? Hint: Each of the 2n possible orderings of red and
n
black cards corresponds to a random walk of length 2n that returns to the
origin at time 2n. Show that between each pair of successive equalizations,
the parent will be right exactly once more than he will be wrong. Explain
why this means that the average number of correct guesses by the parent is
greater than n by exactly one-half the average number of equalizations. Now
define the random variable Xi to be 1 if there is an equalization at time 2i,
and 0 otherwise. Then, among all relevant paths, we have
2n−2i 2i
E(Xi) = P (Xi = 1) = n−i i .
2n
n
Thus, the expected number of equalizations equals
n 1n 2n − 2i 2i
= 2n n−i .
E Xi
n i=1 i
i=1
One can now use generating functions to find the value of the sum.
It should be noted that in a game such as this, a more interesting question
than the one asked above is what is the probability that the parent wins the
game? For this game, this question was answered by D. Zagier.5 He showed
that the probability of winning is asymptotic (for large n) to the quantity
1 √1 .
+
2 22
*11 Prove that 1 n (2n)!
and 42n k!k!(n − k)!(n − k)! ,
u2(2n) =
k=0
u(23n) = 1 (2n)!
62n j!j!k!k!(n − j − k)!(n − j − k)! ,
j,k
4K. Levasseur, “How to Beat Your Kids at Their Own Game,” Mathematics Magazine vol. 61,
no. 5 (December, 1988), pp. 301-305.
5D. Zagier, “How Often Should You Beat Your Kids?” Mathematics Magazine vol. 63, no. 2
(April 1990), pp. 89-92.
486 CHAPTER 12. RANDOM WALKS
where the last sum extends over all non-negative j and k with j + k ≤ n. Also
show that this last expression may be rewritten as
1 2n 1 n! 2
22n n j,k 3n j!k!(n − j − k)! .
*12 Prove that if n ≥ 0, then
n n2 2n
=.
kn
k=0
Hint: Write the sum as nn
n
k n−k
k=0
and explain why this is a coefficient in the product
(1 + x)n(1 + x)n .
Use this, together with Exercise 11, to show that
1 2n n n 2 1 2n 2
42n n k = 42n n .
u2(2n) =
k=0
*13 Using Stirling’s Formula, prove that
2n ∼ √22n .
n πn
*14 Prove that 1 n!
3n j!k!(n − j − k)! = 1 ,
j,k
where the sum extends over all non-negative j and k such that j + k ≤ n.
Hint: Count how many ways one can place n labelled balls in 3 labelled urns.
*15 Using the result proved for the random walk in R3 in Example 12.2, explain
why the probability of an eventual return in Rn is strictly less than one, for
all n ≥ 3. Hint: Consider a random walk in Rn and disregard all but the first
three coordinates of the particle’s position.
12.2 Gambler’s Ruin
In the last section, the simplest kind of symmetric random walk in R1 was studied.
In this section, we remove the assumption that the random walk is symmetric.
Instead, we assume that p and q are non-negative real numbers with p + q = 1, and
that the common distribution function of the jumps of the random walk is
fX (x) = p, if x = 1,
q, if x = −1.
12.2. GAMBLER’S RUIN 487
One can imagine the random walk as representing a sequence of tosses of a weighted
coin, with a head appearing with probability p and a tail appearing with probability
q. An alternative formulation of this situation is that of a gambler playing a sequence
of games against an adversary (sometimes thought of as another person, sometimes
called “the house”) where, in each game, the gambler has probability p of winning.
The Gambler’s Ruin Problem
The above formulation of this type of random walk leads to a problem known as the
Gambler’s Ruin problem. This problem was introduced in Exercise 23, but we will
give the description of the problem again. A gambler starts with a “stake” of size s.
She plays until her capital reaches the value M or the value 0. In the language of
Markov chains, these two values correspond to absorbing states. We are interested
in studying the probability of occurrence of each of these two outcomes.
One can also assume that the gambler is playing against an “infinitely rich”
adversary. In this case, we would say that there is only one absorbing state, namely
when the gambler’s stake is 0. Under this assumption, one can ask for the proba-
bility that the gambler is eventually ruined.
We begin by defining qk to be the probability that the gambler’s stake reaches 0,
i.e., she is ruined, before it reaches M , given that the initial stake is k. We note that
q0 = 1 and qM = 0. The fundamental relationship among the qk’s is the following:
qk = pqk+1 + qqk−1 ,
where 1 ≤ k ≤ M − 1. This holds because if her stake equals k, and she plays one
game, then her stake becomes k + 1 with probability p and k − 1 with probability
q. In the first case, the probability of eventual ruin is qk+1 and in the second case,
it is qk−1. We note that since p + q = 1, we can write the above equation as
p(qk+1 − qk) = q(qk − qk−1) ,
or
q
qk+1 − qk = p (qk − qk−1) .
From this equation, it is easy to see that
qk+1 − qk = q k (12.7)
p
(q1 − q0) .
We now use telescoping sums to obtain an equation in which the only unknown is
q1:
−1 = qM − q0
M −1
= (qk+1 − qk) ,
k=0
488 CHAPTER 12. RANDOM WALKS
so
M −1 q k
p
−1 = (q1 − q0)
k=0
= M −1 q k
p
(q1 − q0) .
k=0
If p = q, then the above expression equals
(q/p)M − 1
(q1 − q0) (q/p) − 1 ,
while if p = q = 1/2, then we obtain the equation
−1 = (q1 − q0)M .
For the moment we shall assume that p = q. Then we have
q1 − q0 = − (q/p) − 1 .
(q/p)M − 1
Now, for any z with 1 ≤ z ≤ M , we have
z−1
qz − q0 = (qk+1 − qk)
k=0
z−1 q k
p
= (q1 − q0)
k=0
= −(q1 − q0) (q/p)z − 1
(q/p) − 1
(q/p)z − 1
= − .
(q/p)M −1
Therefore,
qz = 1 − (q/p)z − 1
(q/p)M − 1
(q/p)M − (q/p)z
= (q/p)M − 1 .
Finally, if p = q = 1/2, it is easy to show that (see Exercise 10)
M −z
qz = M .
We note that both of these formulas hold if z = 0.
We define, for 0 ≤ z ≤ M , the quantity pz to be the probability that the
gambler’s stake reaches M without ever having reached 0. Since the game might
12.2. GAMBLER’S RUIN 489
continue indefinitely, it is not obvious that pz + qz = 1 for all z. However, one can
use the same method as above to show that if p = q, then
(q/p)z − 1
qz = (q/p)M − 1 ,
and if p = q = 1/2, then
z
qz = M .
Thus, for all z, it is the case that pz + qz = 1, so the game ends with probability 1.
Infinitely Rich Adversaries
We now turn to the problem of finding the probability of eventual ruin if the gambler
is playing against an infinitely rich adversary. This probability can be obtained by
letting M go to ∞ in the expression for qz calculated above. If q < p, then the
expression approaches (q/p)z, and if q > p, the expression approaches 1. In the
case p = q = 1/2, we recall that qz = 1 − z/M . Thus, if M → ∞, we see that the
probability of eventual ruin tends to 1.
Historical Remarks
In 1711, De Moivre, in his book De Mesura Sortis, gave an ingenious derivation
of the probability of ruin. The following description of his argument is taken from
David.6 The notation used is as follows: We imagine that there are two players, A
and B, and the probabilities that they win a game are p and q, respectively. The
players start with a and b counters, respectively.
Imagine that each player starts with his counters before him in a pile,
and that nominal values are assigned to the counters in the following
manner. A’s bottom counter is given the nominal value q/p; the next is
given the nominal value (q/p)2, and so on until his top counter which
has the nominal value (q/p)a. B’s top counter is valued (q/p)a+1, and
so on downwards until his bottom counter which is valued (q/p)a+b.
After each game the loser’s top counter is transferred to the top of the
winner’s pile, and it is always the top counter which is staked for the
next game. Then in terms of the nominal values B’s stake is always
q/p times A’s, so that at every game each player’s nominal expectation
is nil. This remains true throughout the play; therefore A’s chance of
winning all B’s counters, multiplied by his nominal gain if he does so,
must equal B’s chance multiplied by B’s nominal gain. Thus,
Pa q a+1 q a+b = Pb q +···+ q a . (12.8)
p pp
p +···+
6F. N. David, Games, Gods and Gambling (London: Griffin, 1962).
490 CHAPTER 12. RANDOM WALKS
Using this equation, together with the fact that
Pa + Pb = 1 ,
it can easily be shown that
(q/p)a − 1
Pa = (q/p)a+b − 1 ,
if p = q, and a
Pa = a + b ,
if p = q = 1/2.
In terms of modern probability theory, de Moivre is changing the values of the
counters to make an unfair game into a fair game, which is called a martingale.
With the new values, the expected fortune of player A (that is, the sum of the
nominal values of his counters) after each play equals his fortune before the play
(and similarly for player B). (For a simpler martingale argument, see Exercise 9.) De
Moivre then uses the fact that when the game ends, it is still fair, thus Equation 12.8
must be true. This fact requires proof, and is one of the central theorems in the
area of martingale theory.
Exercises
1 In the gambler’s ruin problem, assume that the gambler initial stake is 1
dollar, and assume that her probability of success on any one game is p. Let
T be the number of games until 0 is reached (the gambler is ruined). Show
that the generating function for T is
h(z) = 1 − 1 − 4pqz2 ,
2pz
and that q/p, if q ≤ p,
1, if q ≥ p,
h(1) =
and 1/(q − p), if q > p,
∞, if q = p.
h (1) =
Interpret your results in terms of the time T to reach 0. (See also Exam-
ple 10.7.)
√
2 Show that the Taylor series expansion for 1 − x is
√∞ 1/2 xn ,
1−x= n
n=0
where the binomial coefficient 1/2 is
n
1/2 (1/2)(1/2 − 1) · · · (1/2 − n + 1)
=.
n n!
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