Some general strong laws for weighted sums of stochastically dominated random variables. (English) Zbl 0617.60028

Let \(\{Y_ n\), \(n\geq 1\}\) be a sequence of r.v.’s defined on a probability space. Let \(\{\gamma_ n\), \(n\geq 1\}\) be r.v.’s or constants and let \(\{a_ n\), \(n\geq 1\}\) and \(\{b_ n\), \(n\geq 1\}\) be constants with \(b_ n\uparrow \infty\). Then \(\{a_ n(Y_ n-\gamma_ n)\), \(n\geq 1\}\) is said to obey the general strong law of large numbers (SLLN) with norming constants \(\{b_ n\), \(n\geq 1\}\) if the normed weighted sum \(\sum^{\infty}_{1}a_ j(Y_ j-\gamma_ j)/b_ n\) converges almost certainly to 0.
In this paper, the author discusses conditions on \(\{a_ n\}\), \(\{b_ n\}\) and \(\{\gamma_ n\}\) that suffice the SLLN for \(\{a_ n(Y_ n- \gamma_ n)\}\) without assumptions regarding the joint distribution of \(\{Y_ n\}\). He proves two main theorems from which both old and new results for i.i.d. random variables are obtained.
Reviewer: H.Takahata


60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
Full Text: DOI


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