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Journal Article

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

Norbert Herrndorf
The Annals of Probability
Vol. 12, No. 1 (Feb., 1984), pp. 141-153
Stable URL: http://www.jstor.org/stable/2243601
Page Count: 13
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## Abstract

Let (Xn)n ∈ 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 (Xn) 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 |Xn|β for some β ∈ (2, ∞]. For the special cases of strongly mixing sequences (Xn) with $\alpha(k) = \sup \alpha_n(k) = O(k^{-a})$ for some $a > 1$, or α(k) = O(b-k) for some $b > 1$, we obtain functions fβ(n) such that |Xn|β = o(fβ(n)) for some β ∈ (2, ∞] is sufficient for the functional c.l.t., but the c.l.t. may fail to hold if |Xn|β = O(fβ(n)).

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