## Precise asymptotics for a new kind of complete moment convergence.(English)Zbl 1104.60015

Summary: We introduce a new kind of complete moment convergence, which includes complete convergence as a special case. And we also achieve some results about precise asymptotics for this kind of complete moment convergence. For example, let $$\{X, X_n;n\geq 1\}$$ be a sequence of i.i.d. random variables, then $\lim_{\varepsilon \searrow 0}\frac{1}{-\log \varepsilon}\sum^\infty_{n=1}\frac{1}{n^2} ES^2_nI \bigl\{|S_n| \geq\varepsilon n\bigr\}=2\sigma^2$ holds, if and only if $$EX=0$$, $$EX^2=\sigma^2$$ and $$EX^2\log^+|X|<\infty$$.

### MSC:

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

### Keywords:

independent random variables; complete convergence
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### References:

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