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

Abstract

For each k=1,2,⋯k = 1, 2, \cdots let n=n(k)n = n(k), let m=m(k)m = m(k), and suppose yk1,⋯,ykny_1^k, \cdots, y_n^k is an mm-dependent sequence of random variables. We assume the random variables have (2+δ)(2 + \delta)th moments, that m2+2/δ/n→0m^{2 + 2/\delta}/n \rightarrow 0, and other regularity conditions, and prove that n−12(yk1+⋯+ykn)n^{-\frac{1}{2}}(y_1^k + \cdots + y_n^k) is asymptotically normal. An example showing sharpness is given.