On the almost sure convergence, of order \(\alpha\) in the sense of Césaro, \(0<\alpha<1\), for independent and identically distributed random variables. (Q1095489)
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scientific article; zbMATH DE number 4028512
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the almost sure convergence, of order \(\alpha\) in the sense of Césaro, \(0<\alpha<1\), for independent and identically distributed random variables. |
scientific article; zbMATH DE number 4028512 |
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On the almost sure convergence, of order \(\alpha\) in the sense of Césaro, \(0<\alpha<1\), for independent and identically distributed random variables. (English)
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1988
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We prove the following theorem: Given \(0<\alpha \leq 1\), the (C,\(\alpha)\)- means of a sequence of i.i.d. random variables \(X_ n\) converge a.s. iff \(E| X_ n| ^{1/\alpha}<\infty.\) For \(<\alpha \leq 1\) and \(0<\alpha <\) this result is essentially known. We give here a proof of the case \(\alpha =\); an important tool is a theorem of \textit{P. L. Hsu} and \textit{H. Robbins} [Proc. Natl. Acad. Sci. USA 33, 25-31 (1947; Zbl 0030.20101)].
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almost sure convergence
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