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)



Weak laws for dependent sums

Author: William L. Steiger
Journal: Proc. Amer. Math. Soc. 41 (1973), 278-281
MSC: Primary 60F05
MathSciNet review: 0319242
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A general weak law of large numbers for sums $ {S_n} = {X_1} + \cdots + {X_n}$ is proved. That is, without assuming the existence of any moments, and allowing any sort of dependence structure, conditions are given for $ {S_n}/n \to 0$ in probability; the conditions are not necessary. However they are sufficient for a much stronger statement, namely that $ {S_{{\nu _n}}}/{\nu _n} \to 0$ in probability in many cases where positive, integer-valued random variables $ {\nu _n} \to \infty $.

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

  • [1] A. N. Kolmogoroff, Über die Summen durch den Zufall bestimmter unabhängiger Grössen, Math. Ann. 99 (1928), 309-319: Math. Ann. 102 (1929), 484-488. MR 1512451
  • [2] P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris, 1954.
  • [3] J. Mogyorodi, A remark on limiting distribution for sums of a random number of independent random variables, Rev. Roumaine Math. Pures Appl. 16 (1971), 551-557. MR 44 #6011. MR 0288816 (44:6011)
  • [4] P. Révész, The laws of large numbers, Probability and Math. Statist., vol. 4, Academic Press, New York, 1968. MR 39 #6391. MR 0245079 (39:6391)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60F05

Retrieve articles in all journals with MSC: 60F05

Additional Information

Keywords: Weak law of large numbers, dependent random variables, random sums
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society