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

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.