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Convergence and limit theorems for subsequences of random variables

V. F. Gaposhkin

Moscow

Abstract: It is shown that if $X_n$ $(n=1,2,\dots)$ are random variables and $X_n\to0$ weakly in $L_2(\Omega)$, $X_n^2\to1$ weakly in $L_1(\Omega)$ then there exists a subsequence $X_{n_k}$ which is equivalent to $\{Y_k\}$, and $\sum_1^na_kY_k$ is a martingale (see Lemma A).
This fact is used in the rest of the paper to prove some results about subsequences of random variables: in section 2 — convergence and the strong law of large numbers; in section 3 — the central limit theorem; in section 4 — the law of the iterated logarithm.

Received: 12.03.1970

 English version: Theory of Probability and its Applications, 1973, 17:3, 378–400

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