Almost sure stability of partial sums of uniformly bounded random variables

Author:
Theodore P. Hill

Journal:
Proc. Amer. Math. Soc. **89** (1983), 685-690

MSC:
Primary 60F15; Secondary 60G42

MathSciNet review:
718997

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Abstract: Suppose , is a sequence of real numbers with . If a.s. for every sequence of independent nonnegative uniformly bounded random variables , satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables , , a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, , then the assumption of nonnegativity may be dropped.

**[1]**B. M. Brown,*A conditional setting for some theorems associated with the strong law*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**19**(1971), 274–280. MR**0301785****[2]**Y. S. Chow,*Local convergence of martingales and the law of large numbers*, Ann. Math. Statist.**36**(1965), 552–558. MR**0182040****[3]**Lester E. Dubins and David A. Freedman,*A sharper form of the Borel-Cantelli lemma and the strong law*, Ann. Math. Statist.**36**(1965), 800–807. MR**0182041****[4]**T. P. Hill,*Conditional generalizations of strong laws which conclude the partial sums convergence almost surely*, Ann. Probab.**10**(1982), no. 3, 828–831. MR**659552****[5]**P. Levy,*Théorie de l'addition des variables aléatoires*, Gauthier-Villars, Paris, 1937.**[6]**Pál Révész,*The laws of large numbers*, Probability and Mathematical Statistics, Vol. 4, Academic Press, New York-London, 1968. MR**0245079****[7]**William F. Stout,*Almost sure convergence*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 24. MR**0455094**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0718997-9

Keywords:
Almost-sure stability of partial sums,
martingale strong law of large numbers,
conditional generalizations of strong laws

Article copyright:
© Copyright 1983
American Mathematical Society