Asymptotic results for random sums of dependent random variables

Umit Islak

doi:10.1016/j.spl.2015.10.015

Asymptotic results for random sums of dependent random variables

Umit Islak

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Our main result is a central limit theorem for random sums of the form ∑_i=1^Nn X_i, where {X_i}_i≥1 is a stationary m-dependent process and N_n is a random index independent of {X_i}_i≥1. This extends the work of Chen and Shao on the i.i.d. case to a dependent setting and provides a variation of a recent result of Shang on m-dependent sequences. Further, a weak law of large numbers is proven for ∑_i=1^Nn, and the results are exemplified with applications on moving average and descent processes.

Original language	English (US)
Pages (from-to)	22-29
Number of pages	8
Journal	Statistics and Probability Letters
Volume	109
DOIs	https://doi.org/10.1016/j.spl.2015.10.015
State	Published - Feb 1 2016

Keywords

Central limit theorem
Concentration inequality
Local dependence
Random sums
Stein's method

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abstract = "Our main result is a central limit theorem for random sums of the form ∑i=1Nn Xi, where {Xi}i≥1 is a stationary m-dependent process and Nn is a random index independent of {Xi}i≥1. This extends the work of Chen and Shao on the i.i.d. case to a dependent setting and provides a variation of a recent result of Shang on m-dependent sequences. Further, a weak law of large numbers is proven for ∑i=1Nn, and the results are exemplified with applications on moving average and descent processes.",

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AB - Our main result is a central limit theorem for random sums of the form ∑i=1Nn Xi, where {Xi}i≥1 is a stationary m-dependent process and Nn is a random index independent of {Xi}i≥1. This extends the work of Chen and Shao on the i.i.d. case to a dependent setting and provides a variation of a recent result of Shang on m-dependent sequences. Further, a weak law of large numbers is proven for ∑i=1Nn, and the results are exemplified with applications on moving average and descent processes.

KW - Central limit theorem

KW - Concentration inequality

KW - Local dependence

KW - Random sums

KW - Stein's method

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