## The Annals of Mathematical Statistics

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

H. W. Block

#### 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

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

#### 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