A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables

Abstract

Let $(X_n)_{n \in \mathscr{X J}}$ be a sequence of r.v.'s with $E X_n = 0, E(\sum^n_{i = 1} X_i)^2/n \rightarrow \sigma^2 > 0, \sup_{n,m}E(\sum^{m + n}_{i = m + 1} X_i)^2/n < \infty$. We prove the functional c.l.t. for $(X_n)$ under assumptions on $\alpha_n(k) = \sup\{|P(A \cap B) - P(A)P(B)|:A \in \sigma(X_i: 1 \leq i \leq m), B \in \sigma(X_i: m + k \leq i \leq n), 1 \leq m \leq n - k\}$ and the asymptotic behaviour of $\|X_n\|_\beta$ for some $\beta \in (2, \infty\rbrack$. For the special cases of strongly mixing sequences $(X_n)$ with $\alpha(k) = \sup \alpha_n(k) = O(k^{-a})$ for some $a > 1$, or $\alpha(k) = O(b^{-k})$ for some $b > 1$, we obtain functions $f_\beta(n)$ such that $\|X_n\|_\beta = o(f_\beta(n))$ for some $\beta \in (2, \infty\rbrack$ is sufficient for the functional c.l.t., but the c.l.t. may fail to hold if $\|X_n\|_\beta = O(f_\beta(n))$.