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A useful identity in convergence of random series

July 23, 2012 Leave a comment

Let $X_1, X_2,\ldots,$ be independent and let $S_{m,n} = X_{m+1} + \cdots + X_n.$ Then for any $a>0$

$P(\max_{m 2a)\min_{ma).$

The identity can be used to prove the following results:

1) P. Levy: Let $X_1, X_2,\ldots$ be independent and let $S_n = X_1+\cdots+X_n$. If $\lim_{n\rightarrow \infty} S_n$ exists in probability, then it also exists almost surely.

2) Let  $X_1, X_2,\ldots$  be i.i.d. and $S_n = X_1+\cdots+X_n.$ If $\lim_{n\rightarrow \infty} S_n/n \rightarrow 0$ in probability, then $(\max_{1\leq m\leq n} S_m)/n \rightarrow 0$ in probability.

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Categories: Probability Theory