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Strong laws of large numbers for arrays of row-wise independent random variables. (English) Zbl 0685.60032

Let \(\{X_{nk}:\) \(k=1,...,n\}\), \(n\geq 1\) be a triangular array of row- wise independent r.v’s with E \(X_{nk}=0\) \(\forall k\), \(\forall n\). The authors assume that there is a r.v. X with E \(| X|^{2p}<\infty\), \(1\leq p<2\) and such that \(P(| X_{nk}| >t)\leq P(| X| >t)\). The main result states that for \(n\to \infty\) \[ S_ n=1/n^{1/p}\sum^{n}_{k=1}X_{nk}\to 0\quad completely, \] in the sense that \(\sum^{\infty}_{n=1}P(| S_ n| >\epsilon)<\infty\). By virtue of the Borel-Cantelli lemma complete convergence implies a.s. convergence. Thus, the authors’ main result is a SLLN for arrays of not identically distributed independent r.v’s.
Reviewer: E.Pancheva

MSC:

60F15 Strong limit theorems
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