Weak laws for dependent sums
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- by William L. Steiger PDF
- Proc. Amer. Math. Soc. 41 (1973), 278-281 Request permission
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
- A. Kolmogoroff, Über die Summen durch den Zufall bestimmter unabhängiger Größen, Math. Ann. 99 (1928), no. 1, 309–319 (German). MR 1512451, DOI 10.1007/BF01459098 P. Lévy, Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris, 1954.
- J. Mogyoródi, A remark on limiting distributions for sums of a random number of independent random variables, Rev. Roumaine Math. Pures Appl. 16 (1971), 551–557. MR 288816
- Pál Révész, The laws of large numbers, Probability and Mathematical Statistics, Vol. 4, Academic Press, New York-London, 1968. MR 0245079
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 278-281
- MSC: Primary 60F05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0319242-2
- MathSciNet review: 0319242