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A two-sided estimate in the Hsu-Robbins-Erdős law of large numbers. (English) Zbl 0911.60021

From the author’s abstract: Let \(X_1,X_2,\ldots \) be independent identically distributed random variables. Then, P. L. Hsu and H. Robbins [Proc. Natl. Acad. Sci. USA 33, 25-31 (1947; Zbl 0030.20101)] together with P. Erdős [Ann. Math. Stat. 20, 286-291 (1949; Zbl 0033.29001) and ibid. 21, 138 (1950; Zbl 0035.21403)] have proved that \[ S(\lambda) \overset \text{def} =P(| X_1+\ldots +X_n| \geq \lambda n) < \infty , \quad \forall \;\lambda >0, \] if and only if \(E[X^2_1]<\infty \) and \(E[X_1]=0\). We prove that there are absolute constants \(C_1\), \(C_2 \in (0,\infty)\) such that if \(X_1,X_2,\ldots \) are independent identically distributed mean zero random variables, then \[ C_1 \lambda ^{-2} E[X^2_1 \cdot 1_{\{| X_1| \geq \lambda \}}] \leq S(\lambda) \leq C_2 \lambda ^{-2} E[X^2_1\cdot 1_{\{| X_1| \geq \lambda \}}], \] for every \(\lambda >0\).
Reviewer: P.Lachout (Praha)

MSC:

60F15 Strong limit theorems
60F99 Limit theorems in probability theory
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