MathDB

10. Let

X_1, X_2, \dots

be independent random variables with the same distribution

P(X_i = 1) = P(X_i = -1)=\frac{1}{2}\qquad (i= 1, 2, \dots )

Define

S_0=0, Sn=X_1 +X_2+\dots +X_n \qquad (n=1, 2, \dots

\xi (x,n) = \left | \{k : 0 \leq k \leq n, S_k= x \} \right |\qquad (x=0, \pm 1, \pm 2, \dots

),
and

\alpha(n)= \left | \{ x: \xi(x,n)=a \} \right |\qquad (n=0,1,\dots

).
Prove that

P(\lim \inf \alpha(n)=0) =1

and that there is a number

0<c<\infty

such that

P(\lim \inf \alpha(n)/\log n=c) =1

(P.24) [P. Révész]

Problem Statement