## The Annals of Probability

### A Central Limit Theorem for $m$-Dependent Random Variables with Unbounded $m$

Kenneth N. Berk

#### Abstract

For each $k = 1, 2, \cdots$ let $n = n(k)$, let $m = m(k)$, and suppose $y_1^k, \cdots, y_n^k$ is an $m$-dependent sequence of random variables. We assume the random variables have $(2 + \delta)$th moments, that $m^{2 + 2/\delta}/n \rightarrow 0$, and other regularity conditions, and prove that $n^{-\frac{1}{2}}(y_1^k + \cdots + y_n^k)$ is asymptotically normal. An example showing sharpness is given.

#### Article information

Source
Ann. Probab. Volume 1, Number 2 (1973), 352-354.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176996992

Digital Object Identifier
doi:10.1214/aop/1176996992

Mathematical Reviews number (MathSciNet)
MR350815

Zentralblatt MATH identifier
0263.60006

JSTOR
Berk, Kenneth N. A Central Limit Theorem for $m$-Dependent Random Variables with Unbounded $m$. Ann. Probab. 1 (1973), no. 2, 352--354. doi:10.1214/aop/1176996992. http://projecteuclid.org/euclid.aop/1176996992.