The Annals of Mathematical Statistics

Accuracy of Convergence of Sums of Dependent Random Variables with Variances Not Necessarily Finite

H. W. Block

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Abstract

Let $S_n = \sum^{k_n}_{k=1} X_{nk}$ and $X$ be random variables with distribution functions $F_n(x)$ and $F(x)$. No assumptions are made that the $(X_{nk})$ have finite means or variances. Also, no independence conditions are assumed. A bound is found for $M_n = \sup_{-\infty<x<\infty}|F_n(x) - F(x)|.$ This bound involves various truncated moments and conditional probabilities and expectations. A typical quantity involved is $\sum^{k_n}_{k=1} E|E (X_{n^k}|\sum^{k-1}_{j=1} X_{nj}) - E(X_{n^k})|$. Using this bound, particular conditions are found so that $S_n$ converges in distribution to $X$.

Article information

Source
Ann. Math. Statist., Volume 42, Number 6 (1971), 2134-2138.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177693080

Digital Object Identifier
doi:10.1214/aoms/1177693080

Mathematical Reviews number (MathSciNet)
MR298733

Zentralblatt MATH identifier
0227.60019

JSTOR
links.jstor.org

Citation

Block, H. W. Accuracy of Convergence of Sums of Dependent Random Variables with Variances Not Necessarily Finite. Ann. Math. Statist. 42 (1971), no. 6, 2134--2138. doi:10.1214/aoms/1177693080. https://projecteuclid.org/euclid.aoms/1177693080


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