## 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

### MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks
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### References:

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