Weighted sums of random variables attracted to stable laws (Q1056987)

Let \(X_ k\), \(k=1,2,...\), be a sequence of independent random variables. Assume that \(X_ k\) belongs to the domain of normal attraction of the stable law \(G_{\alpha}(\cdot,\lambda_ k,\beta_ k)\), where \(0<\alpha \leq 2\), \(| \beta_ k| \leq 1\) and the scale parameter \(\lambda_ k>0\). Further let \(a_ k(v)\), \(k=1,2,...\), be a nonnegative sequence of discounting factors depending on a parameter \(v\in (0,1)\) and satisfying the conditions \(0<B_ v^{\alpha}=\sum^{\infty}_{k=0}a_ k(v)\lambda_ k\) if \(0<v<1\) and \(B_ v\to \infty\) as \(v\to 1-0.\) The aim of the paper is to determine conditions for the almost sure convergence of the series \(Z_ v=\sum^{\infty}_{k=0}a_ k(v)X_ k\) if \(0<v<1\) and for the rate of convergence of the distribution function of \(S_ v=B_ v^{-1}(Z_ v-A_ v)\) as \(v\to 1-0\) to a stable law, where \(A_ n\) is a certain constant.