A strong law for weighted sums of i.i.d. random variables. (English) Zbl 0833.60031

Let \(X\), \(X_1, X_2, \dots, X_n, \dots\) be i.i.d. random variables with \({\mathbf E}X = 0\) and \(S_n = \sum^n_{i = 1} a_{in} X_n\), where \(\{a_{in}\}_{i = 1(1)n}\) is an array of constants. E.g., for the sequence \(\{S_n\}\) of weighted sums a strong law is proved in the case \(\sup_{n} \root q \of {{1\over n} \sum^n_{i = 1} |a_{in}|^q} < \infty, q\in (1;\infty],\) in the following manner: \[ {\mathbf E} |X|^p < \infty,\;p := {q\over q -1}, \text{ implies } {1\over n} S_n \to 0 \text{ a.s. (Theorem 1.1).} \] Also the case \(q = 1\) is investigated. Extensions to more general normalizing sequences are given and necessary and sufficient conditions are discussed. Several well-known results are contained, e.g. in the case \(q = \infty\), i.e. \(\text{sup} |a_{in}|< \infty\) and \(p = 1\), the result by B. D. Choi and S. H. Sung [Stochastic Anal. Appl. 5, 365-377 (1987; Zbl 0633.60049)].
Reviewer: L.Paditz (Dresden)


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


Zbl 0633.60049
Full Text: DOI


[1] Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987).Regular Variation. Cambridge University Press, Cambridge. · Zbl 0617.26001
[2] Choi, B. D., and Sung, S. H. (1987). Almost sure convergence theorems of weighted sums of random variables.Stochastic Analysis and Applications 5, 365–377. · Zbl 0633.60049
[3] Chow, Y. S. (1966). Some convergence theorems for independent random variables.Ann. Math. Statist. 37, 1482–1493. · Zbl 0152.16905
[4] Chow, Y. S., and Lai, T. L. (1973). Limiting behavior of weighted sums of independent random variables.Ann. Prob. 1, 810–824. · Zbl 0303.60025
[5] Chow, Y. S., and Teicher, H. (1988).Probability Theory: Independence, Interchangeability, Martingales. Springer, New York. · Zbl 0652.60001
[6] Feller, W. (1946). A limit theorem for random variables with infinite moments.Amer. J. Math. 68, 257–262. · Zbl 0060.28704
[7] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables.J. Amer. Statist. Assoc. 58, 13–30. · Zbl 0127.10602
[8] Hsu, P. L., and Robbins, H. (1947). Complete convergence and the law of large numbers.Proc. Nat. Acad. Sci. USA 33, 25–31. · Zbl 0030.20101
[9] Klass, M., and Teicher, H. (1977). Iterated logarithm laws for asymmetric random variables barely with or without finite mean.Ann. Prob. 5, 861–874. · Zbl 0372.60042
[10] Rosalsky, A. (1993). On the almost certain limiting behavior of normed sums of identically distributed positive random variables.Statistics & Probability Letters 16, 65–70. · Zbl 0765.60020
[11] Sawyer, S. (1966). Maximal inequalities of weak type.Ann. Math. 84, 157–174. · Zbl 0186.20503
[12] Stout, W. F. (1968). Some results on the complete and almost sure convergence of linear combinations of independent random variables and martingale differences.Ann. Math. Statist. 39, 1549–1562. · Zbl 0165.52702
[13] Stout, W. F. (1974).Almost Sure Convergence. Academic Press, New York. · Zbl 0321.60022
[14] Teicher, H. (1985). Almost certain convergence in double arrays.Z. Wahrsch. Verw. Geb. 69, 331–345. · Zbl 0548.60028
[15] Thrum, R. (1987). A remark on almost sure convergence of weighted sums.Prob. Th. Rel. Fields 75, 425–430. · Zbl 0599.60031
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