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Functional limit theorems for weighted sums of i.i.d. random variables. (English) Zbl 0567.60037

Let \(\{X_ i\}\) be a sequence of independent and identically distributed random variables belonging to the domain of attraction of a stable law with index \(\alpha\), \(0<\alpha \leq 2\). The paper presents some functional limit theorems of the processes \(t\to \sum^{\infty}_{i=0}c_ i(\lambda; t)X_ i\) for some weights \(c_ i(\lambda; t)\), as \(\lambda\) \(\to \infty\). The basic idea of the approach to the problem is to use the so-called point-process method. Applications to the special weights concerning the classical summability methods such as Abel, Borel and Euler are given.

MSC:

60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
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