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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
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