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The invariance principle for $$\phi$$-mixing sequences. (English) Zbl 0626.60031
Let $$\{X_ n:n\geq 1\}$$ be a $$\phi$$-mixing sequence of centered random variables with finite variance. Denote the sum of $$X_ 1$$ through $$X_ n$$ by S(n). Suppose $$ES^ 2(n)\to \infty$$ as $$n\to \infty$$. Let $$k_ n$$ be an increasing sequence of positive numbers for which $$k_ 0=0$$, and $$\max \{k_ i-k_{i-1}:1\leq i\leq n\}=o(k_ n)$$. Assume $$ES^ 2(n)=k_ nh(k_ n)$$ for a slowly varying function h, and that $$\max \{| X_ i|:1\leq i\leq n\}/ES^ 2(n)\to 0$$ in probability. Set $$m_ n(t)=\max \{i:k_ i\leq tk_ n\}$$ and $$Y_ n(t)=S(m_ n(t))/ES^ 2(n).$$
Under a uniform integrability condition on the increments of $$Y_ n(t)$$, this paper shows that $$Y_ n$$ converge weakly to a Wiener process. N. Herrndorf [Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 97-108 (1983; Zbl 0494.60036)] established this result when $$k_ n=n$$. This paper allows for a variable “clock speed” in the definition of $$Y_ n$$.
Reviewer: A.R.Dabrowski

MSC:
 60F17 Functional limit theorems; invariance principles