Strong Theorems on Coin Tossing

Proceedings of the International Congress of Mathematicians
Helsinki, 1978

Strong Theorems on Coin Tossing

P. Révész

I. The number of heads in short blocks.

1.1. The length of the longest head run. The length of the longest head run in a coin

tossing sequence was investigated in the very early days of probability theory (see

e.g. [10]). A teaching experiment of T. Varga originated a new attack on this problem.

His experiment goes like this:

A class of school children is divided into two sections. In one of the sections each

child is given a coin which they throw two hundred times, recording the resulting

head and tail sequence on a piece of paper. In the other section the children do not

receive coins, but are told instead that they should try to write down a "random"

head and tail sequence of length two hundred. Collecting these slips of paper, he

then tries to subdivide them into their original groups. Most of the times he succeeds

quite well. His secret is that he had observed that in a randomly produced sequence

of length two hundred, there are, say, head-runs of length seven. On the other

hand, he had also observed that most of those children who were to wiite down an

imaginary random sequence are usually afraid of writing down runs of longer than

four. Hence, in order to find the slips coming from the coin tossing group, he simply

selects the ones which contain runs longer than five.

This experiment led T. Varga to ask: What is the length of the longest run of

pure heads in n Bernoulli trials?

Introduce the following notations:

Let Xl9 Z2,... be a sequence of independent and identically distributed random
variables with P(X1=0)=P(X1=l) = l/29 let S0=0, Sn=X1+Z2+...+Ar„
(«=1,2,...) and

I(N9K)=QjmxK(Sn+K--Sn) (N^K).

750 P. Révész

Define the r.v.'s ZN (N=l9 2,...) as follows: let ZN be the largest integer for
which I(N9ZN)=ZN. This Zn is the length of the longest head-run.

A characterization of the behaviour of Z„ was given by Erdôs and Révész. It is
presented in this section.

THEOREM 1.1 [7]. Let s be any positive number. Then for almost all co£Q (the
basic space) there exists a finite NQ=N0(CO9ë) such that1

ZN ^ [log JV—log log log ÌV+log loge—2—ß] = ai(A0 = o^
ifN^N0.

This result is quite near to the best possible one in the following sense :

THEOREM 1.2 [7]. Let e be any positive number. Then for almost all co£Q there
exists an infinite sequence Ni=Ni(co9 c) ( J = 1 , 2,...) of integers such that

Zni < [log Nt - log log log Nt+log log e - 1 + e ] = a2(N) = a2.

Theorems 1.1 and 1.2 together say that the length of the longest head-run is larger
than <*! but in general not larger than a2 <;<*!+2. Clearly enough, for some N the
length of the longest head-run can be larger than a2. In our next theorems the lar-
gest possible values of ZN are investigated.

THEOREM 1.3 [7]. Let (a3(w)} be a sequence of positive numbers for which

2'2-a3^=co.

«=1

Then for almost all (D£Q there exists an infinite sequence N^N^co, {a3(«)})
(/= 1,2,...) of integers such that

ZNi ^ a.W)-

This result is the best possible in the following sense:

THEOREM 1.4 [7]. Let {a4(«)} be a sequence of positive numbers for which

Then for almost all co£Q there exists a positive integer N0=N0(co9 {a4(«)}) suchthat

ZN < a4(JV)

if N^NQ.

1.2. The length of blocks containing at most T tails. Theorems 1.1-1.4 are charac-
terizing the length of the longest run containing no tails at all. One can also ask
similar questions about the length of the longest run containing at most T tails.

1 Here and in what follows log means logarithm with base 2; [x] is the integral part of x.

Strong Theorems on Coin Tossing 751

In order to formulate our results, we introduce the following notations: Let Z^(T)
be the largest integer for which

/(i\r,zN(D)>zw(7Vr.

This ZN(T) is the length of the longest run containing at most T tails. Theorems
1.1-1.4 can be generalized as follows:

THEOREM 1.1* [7]. Let s be any positive number. Then for almost all ςQ there
exists a finite N0=N0(co9 T9e) such that

ZN(T) >> [logN+TloglogN-logloglogN-log (Tl) +loglog e-2-e] = ax(i\T, T)

if N^N0.
THEOREM 1.2* [7]. Let e be any positive number. Then for almost all co£Q there

exists an infinite sequence Ni=Nt(co9 T9 e) of integers such that

ZNt < [logiVr+rioglogiVr-logloglog^-log(r!)+logloge-l+e] = a2(AT, T).

THEOREM 1.3* [7]. Let {aa(N9 T)}^^ be a sequence of positive integers for which

Then for almost all co 6 Q there exists an infinite sequence Ni=Ni(œ9 T9 {ua(N9 T)})
of integers such that

ZNi(T) ^ *Q(Ni9 T).

THEOREM 1.4* [7]. Let {^(N9T)}^=1 be a sequence ofpositive integers for which

2(**(N9T))T*'

N=l

Then for almost all co£Q there exists a positive integer N0=N0(co9 T9 {^(N9 T)})
such that

ZN(T) < «t(N9 T)
if N^N0.

1.3. A result ofErdos and Rényi. Erdös and Rényi were the first ones who inves-
tigated the problem proposed by T. Varga, proving strong theorems. One of their
results goes like this:

THEOREM 1.5 [6]. We have

N^OO clogN 2v '

752 P. Révész

with probability 1 where a(l) = l and, if o l then it is the only solution of the
equation

with
h(x) =—xlogx—(l—x)log(l-x) (0 < x < 1).

77?e herewith defined a(-) wo decreasing function with

lim a(c) = 1 «m/ lim a(c) = 0.

CSU ' C^foa v '

1.4. T/ze number of head runs. In Theorems 1.1 and 1.2 we have seen that for all

N9 big enough, there exists a block of size o^(N) containing only heads but it is
not true with <x,2(N). Now we ask what is the number of disjoint blocks of size
ai(iV) containing only heads.

Let vN(k) be the number of blocks of size k (in the interval [0, N]) containing
only heads, that is to say vN(k)=j if there exists a sequence 0</1</1+/c<</2<
l2+k*^...^lj<lj+k^n suchthat

Sìi+k-Sh = k(i = l929...9j)

but

sm+k-sm^k if h+k^m^lw (i==l929...9j).

At first we study the case k=a1(N) and prove 0^c1=c1(s)^c2=c2(s)<oo

THEOREM 1.6. For any 8>0 there exist constants
such that

C1l = Htfm-~M logÄlog N ^ Ktfm-~ supl;o*gfolo(igVfNf = c2 2
with probability 1.

This theorem says that in the interval [09N] there are 0(loglogAO blocks
of size OL^N) containing only heads. This fact is quite surprising, knowing that it

does happen for infinitely many N that there is not any block of size a1(N)+2>>
a.2(N) containing only heads.

Theorem 1.2 also implies that
lim inf vN(kN) = 0

with probability 1 if kN>>a,2(N). However, Theorems 1.3 and 1.4 imply that

x f== °0 Ìiff (5>0,
limsupviV([logiv'+(l + 5)loglogiV])|^1 .f

Now we are interested in

lim sup Vjy([log N+log log N])

and formulate our

Strong Theorems on Coin Tossing 753

THEOREM 1.7. We have
lim sup Vjv([log N+ log log N]) ^ 2

with probability 1.

II. The number of heads in longer blocks.

ILL The most irregular blocks. As we have already remarked, Theorem 1.5
stated that for any c>0 there is a block of size [c log TV] whose density is more
than 1/2. This Theorem also says that in any block of size aN»logN the density
of heads is equal to 1/2. That is to say we have

A CONSEQUENCE OF THEOREM 1.5.

lira ZWM = 1/2

w-~ aN
with probability I, provided that

lim % - -

AT-«, logN

Clearly, in order to get more precise results, we have to investigate the properties
of the sequence

J(N9aN) = 2I(N9aN)-aN.

Introduce the following notations :

7, = 2 ^ - 1 , rn = Ì 7 i = 2S„-n,

f=i

J(N, K) = max (Tn+K-T„) = 2I(N, K)-K,

JHN9K)=QmIx_K\Tn+k^n

J+(N9K)= max max \Tn+l-T±

The limiting behaviour of these sequences was studied in [3], where the main
result says:

THEOREM 2.1 [3]. Let {aN}^=1 be a nondecreasing sequence of positive integers
for which

(i) 0^aN^N(N=l929...)9
(ii) aN/N is nonincreasing9

(iii) l i m ^ o » [fljv/log N] = ~ .

Then lim sup ßNJ(N9 aN) = lim sup ßNJ+ (N9 aN)

yy-+• oo jy-*- oo

= lim sup ßNJ(N,aN)

JV-*-oo

= Urn sup ßNJ+(N9aN) = 1

754 P. Révész: Strong Theorems on Coin Tossing

with probability 1, where

ßN = (2aN)[ln N/aN+\n In N])~1/2.

If we also have

(iv) lim^oo [In N/aNfln In N~\ = ~

then
lim ß„J(N9 aN) = lim ßNJ+(N9 aN)

= lim ßNJ(N, aK) = lim ßN J+ (JV, % ) = 1

N-*- oo N-*• oo

with probability 1.

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