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The weak law of large numbers for nonnegative summands

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 February 2019

Eugene Seneta
Eugene Seneta*
Affiliation:
University of Sydney
*
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: [email protected]
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Abstract

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Khintchine's (necessary and sufficient) slowly varying function condition for the weak law of large numbers (WLLN) for the sum of n nonnegative, independent and identically distributed random variables is used as an overarching (sufficient) condition for the case that the number of summands is more generally [cn],cn→∞. Either the norming sequence {an},an→∞, or the number of summands sequence {cn}, can be chosen arbitrarily. This theorem generalizes results from a motivating branching process setting in which Khintchine's sufficient condition is automatically satisfied. A second theorem shows that Khintchine's condition is necessary for the generalized WLLN when it holds with cn→∞ and an→∞. Theorem 3, which is known, gives a necessary and sufficient condition for Khintchine's WLLN to hold with cn=n and an a specific function of n; it is extended to general cn subject to a growth restriction in Theorem 4. Section 6 returns to the branching process setting.

Keywords

Khintchine's conditionnonnegative summandslowly varying functionarbitrary norming or number of summandssimple branching process

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 241 - 252
Copyright
Copyright © Applied Probability Trust 2018 

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