Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $ {a_1}$, $ {a_2}, \ldots $ is a sequence of real numbers with $ {a_n} \to \infty $. If $ ({X_1} + \cdots + {X_n})/{a_n} = \alpha $ a.s. for every sequence of independent nonnegative uniformly bounded random variables $ {X_1}$, $ {X_2}, \ldots $ satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables $ {Y_1}$, $ {Y_2}, \ldots ,$, $ \lim {\text{ sup}}({Y_1} + \cdots + {Y_n})/{a_n} = \alpha $ a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, $ \Sigma ^\infty a_n^{ - 2} < \infty $, then the assumption of nonnegativity may be dropped.


References [Enhancements On Off] (What's this?)

  • [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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60F15, 60G42

Retrieve articles in all journals with MSC: 60F15, 60G42


Additional Information

DOI: http://dx.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