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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

60F17 Functional limit theorems; invariance principles