Pruss, Alexander R. A two-sided estimate in the Hsu-Robbins-Erdős law of large numbers. (English) Zbl 0911.60021 Stochastic Processes Appl. 70, No. 2, 173-180 (1997). 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) Cited in 2 ReviewsCited in 8 Documents MSC: 60F15 Strong limit theorems 60F99 Limit theorems in probability theory Keywords:rates of convergence; complete convergence; Hsu-Robbins-Erdős law of large numbers; tail probabilities of sums of independent identically distributed random variables Citations:Zbl 0030.20101; Zbl 0033.29001; Zbl 0035.21403 PDFBibTeX XMLCite \textit{A. R. Pruss}, Stochastic Processes Appl. 70, No. 2, 173--180 (1997; Zbl 0911.60021) Full Text: DOI References: [1] Chow, Y. S.; Lai, T. L., Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings, Trans. Amer. Math. Soc., 208, 51-72 (1975) · Zbl 0335.60021 [2] Chow, Y. S.; Lai, T. L., Paley-type inequalities and convergence rates related to the law of large numbers and extended renewal theory, Z. Wahrsch. Verw. Geb., 45, 1-19 (1978) · Zbl 0397.60030 [3] Erdös, P., On a theorem of Hsu and Robbins, Ann. Math. Statist., 20, 286-291 (1949) · Zbl 0033.29001 [4] Erdös, P., Remark on my paper “On a theorem of Hsu and Robbins”, Ann. Math. Statist., 21, 138 (1950) · Zbl 0035.21403 [5] Heyde, C. C., A supplement to the strong law of large numbers, J. Appl. Probab., 12, 173-175 (1975) · Zbl 0305.60008 [6] Hsu, P. L.; Robbins, H., Complete convergence and the law of large numbers, (Proc. Nat. Acad. Sci. USA, 33 (1947)), 25-31 · Zbl 0030.20101 [7] Li, D.; Rao, M. B.; Jiang, T.; Wang, X., Convergence and almost sure convergence of weighted sums of random variables, J. Theoret. Probab., 8, 49-76 (1995) · Zbl 0814.60026 [8] Petrov, V. V., Limit theorems for sums of independent random variables (1987), Nauka: Nauka Moscow, (in Russian) · Zbl 0621.60022 [9] Pruss, A. R., On Spǎtaru’s extension of the Hsu-Robbins-Erdös law of large numbers, J. Math. Anal Appl., 199, 558-576 (1996) · Zbl 0853.60042 [10] Spǎtaru, A., Strengthening the Hsu-Robbins-Erdös theorem, Rev. Roumaine Math. Pur. Appl., 35, 463-465 (1990) · Zbl 0733.60051 [11] Tucker, H. G., A graduate course in probability (1967), Academic Press: Academic Press New York · Zbl 0159.45702 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.