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Weighted sums for i. i. d. random variables with relatively thin tails. (English) Zbl 0956.60013

Strong limit theorems are proved for a sequence of weighted sums \(T_n= \sum^\infty_{k=1} a_{nk}X_k\) \((n= 1,2,\dots)\), where \((a_{nk})_{n,k= 1,2,\dots}\) is a double array satisfying \(\sum^\infty_{k=1} a_{nk}< \infty\) for each \(n\), and where \((X_k)_{k= 1,2,\dots}\) is an i.i.d. sequence with \(EX_1= 0\), and \(E\exp(t|X_1|)^{1/p})< \infty\) in a neighbourhood of \(t= 0\), for some \(p>0\). For special choices of weights, necessity of moment conditions can also be obtained. Various strong laws, based on summability methods, and their convergence rates are discussed including Cesàro, Euler and random walk methods as well as Valiron and running means. The results complement earlier work of Lanzinger (1995, 1998) and Li et al. (1995).

MSC:

60F15 Strong limit theorems
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
60G50 Sums of independent random variables; random walks
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