Geodesic
Convergence and limit theorems for subsequences of random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 401-423

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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. 
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     author = {V. F. Gaposhkin},
     title = {Convergence and limit theorems for subsequences of random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {401--423},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1972_17_3_a0/}
}
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V. F. Gaposhkin. Convergence and limit theorems for subsequences of random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 401-423. http://geodesic.mathdoc.fr/item/TVP_1972_17_3_a0/