MathDB
Miklos Schweitzer 1976_11

Source:

December 30, 2008
logarithmsprobabilitygeometrygeometric transformationreflectionprobability and stats

Problem Statement

Let ξ1,ξ2,... \xi_1,\xi_2,... be independent, identically distributed random variables with distribution P(ξ1=−1)=P(ξ1=1)=12. P(\xi_1=-1)=P(\xi_1=1)=\frac 12 . Write Sn=ξ1+ξ2+...+ξn  (n=1,2,...),   S0=0 , S_n=\xi_1+\xi_2+...+\xi_n \;(n=1,2,...),\ \;S_0=0\ , and Tn=1nmax⁡0≤k≤nSk. T_n= \frac{1}{\sqrt{n}} \max _{ 0 \leq k \leq n}S_k . Prove that lim inf⁡n→∞(log⁡n)Tn=0 \liminf_{n \rightarrow \infty} (\log n)T_n=0 with probability one. P. Revesz
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