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Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass

Avena, Luca; da Costa, Conrado

Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass Thumbnail


Authors

Luca Avena



Abstract

We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples.

Citation

Avena, L., & da Costa, C. (2024). Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass. Journal of Theoretical Probability, 37(1), 81-120. https://doi.org/10.1007/s10959-023-01296-z

Journal Article Type Article
Acceptance Date Sep 29, 2023
Online Publication Date Nov 1, 2023
Publication Date Mar 1, 2024
Deposit Date Nov 7, 2023
Publicly Available Date Nov 7, 2023
Journal Journal of Theoretical Probability
Print ISSN 0894-9840
Electronic ISSN 1572-9230
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 37
Issue 1
Pages 81-120
DOI https://doi.org/10.1007/s10959-023-01296-z
Keywords Toeplitz matrices, 60G50, Law of large numbers, Weighted sums of independent random variables, 60F15
Public URL https://durham-repository.worktribe.com/output/1874194

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