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Complete convergence of weighted sums of martingale differences. (English) Zbl 0698.60035
A sequence \(\{U_ n\), \(n\geq 1\}\) converges completely to the constant c if \[ \sum P(| U_ n-c| >\epsilon)<\infty \quad for\quad every\quad \epsilon >0. \] The now classical result of P. L. Hsu and H. Robbins [Proc. Nat. Acad. Sci. USA 33, 25-31 (1947; Zbl 0030.20101)] concerning complete convergence of the arithmetic mean of i.i.d. random variables has been generalized in various ways. The present paper deals with weighted sums of martingale differences.
Reviewer: A.Gut

MSC:
60F15 Strong limit theorems
60G42 Martingales with discrete parameter
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[1] Chow, Y. S. (1966). Some convergence theorems for independent random variables.Ann. Math. Stat. 37, 1482-1493. · Zbl 0152.16905 · doi:10.1214/aoms/1177699140
[2] Chow, Y. S. and Teicher, H. (1978).Probability Theory, Springer-Verlag, New York. · Zbl 0399.60001
[3] Teicher, H. (1985). Almost certain convergence in double arrays.Z. Wahrsch. verw. Geb. 69, 331-345. · Zbl 0548.60028 · doi:10.1007/BF00532738
[4] Thrum, R. (1987). A remark on almost sure convergence of weighted sums.Prob. Theory Rel. Fields 75, 425-430. · Zbl 0599.60031 · doi:10.1007/BF00318709
[5] Yu, K. F. (1987). On the uniform integrability and almost sure convergence of weighted sums. University of South Carolina Technical Report No. 132.
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