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probability

Source: miklos schweitzer 1998 q10

September 19, 2021
probability and statsprobability

Problem Statement

Let ξ1,ξ2,...\xi_1 , \xi_2 , ... be a series of independent, zero-expected-value random variables for which lim⁡n→∞E(ξn2)=0\lim_{n\to\infty} E(\xi_n ^ 2) = 0, and Sn=∑j=1nξjS_n = \sum_{j = 1}^n \xi_j . Denote by I(A) the indicator function of event A. Prove that 1log⁡n∑k=1n1kI(max⁡1≤j≤k∣Sj∣>k)→0\frac{1}{\log n} \sum_{k = 1}^n \frac1k I\bigg(\max_{1\leq j\leq k} |S_j|>\sqrt k\bigg) \to 0 with probability 1 if n→∞n\to\infty .
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