# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.