Central limit theorem and the law of large numbers in the mean

V. M. Kruglov

Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 824-827

Résumé

Let $\{\xi_{n1},\xi_{n2},\dots,\xi_{nk_n}\}_{n=1}^{\infty}$ be a sequence of independent (for every $n\ge 1$) infinitesimal random variables. We prove that $$ \lim_{n\to\infty}\mathbf P\biggl(\sum_{j=1}^{k_n}\xi_{nj}-A_n<x\biggr)= (2\pi)^{-1/2}\int_{-\infty}^x e^{-u^2/2}\,du $$ for some constants $A_n$, $n=1,2,\dots$, and $$ \lim_{n\to\infty}\mathbf M\biggl|\sum_{j=1}^{k_n}\xi_{nj}-A_n\biggr|^{2q}= (2\pi)^{-1/2}\int_{-\infty}^{\infty}|u|^{2q} e^{-u^2/2}\,du $$ for some $q>0$ if and only if $$ \lim_{n\to\infty}\mathbf M\biggl|\sum_{j=1}^{k_n}\biggl(\xi_{nj}- \mathbf M\biggl\{\xi_{nj}\biggl||\xi_{nj}|<1\biggr\}\biggr)^2-1\biggr|^q=0. $$