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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<j\leq n} |S_{m,j}| > 2a)\min_{m<k\leq n} P(|S_{k,n}|\leq a) \leq P(|S_{m,n}|>a).

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