# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A limit theorem for measurable random processes and its applicationsHTML articles powered by AMS MathViewer

by L. Š. Grinblat
Proc. Amer. Math. Soc. 61 (1976), 371-376 Request permission

## Abstract:

Let the measurable random processes ${\xi _1}(t), \ldots ,{\xi _n}(t), \ldots$ and $\xi (t)$ be defined on $[0,\;1]$. There exists $C$ such that for all $n$ and $t$ we have $E|{\xi _n}(t){|^p} \leqslant C,\;p \geqslant 1$. The following assertion is valid: if for any finite set of points ${t_1}, \ldots ,{t_k} \subset [0,\;1]$ the joint distribution of ${\xi _n}({t_1}), \ldots ,{\xi _n}({t_k})$ converges to the joint distribution of $\xi ({t_1}), \ldots ,\xi ({t_k})$, and if $E|{\xi _n}(t){|^p} \to E|\xi (t){|^p}$ for all $t \in [0,\;1]$, then for any continuous functional $f$ on ${L_p}[0,\;1]$ the distribution of $f({\xi _n}(t))$ converges to the distribution of $f(\xi (t))$. This statement immediately implies the convergence of distributions in some limit theorems for the sums of independent random variables (for example, in one of the theorems of P. Erdös and M. Kac) and in some statistical criteria (for example, in the ${\omega ^2}$-criterion of Cramér and von Mises).
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