Asymptotic results for random sums of dependent random variables

Ümit Işlak

doi:10.1016/j.spl.2015.10.015

Asymptotic results for random sums of dependent random variables

Ümit Işlak

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

11 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
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2015 Elsevier B.V.

Keywords

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

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10.1016/j.spl.2015.10.015

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title = "Asymptotic results for random sums of dependent random variables",

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.",

keywords = "Central limit theorem, Concentration inequality, Local dependence, Random sums, Stein's method",

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year = "2016",

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doi = "10.1016/j.spl.2015.10.015",

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T1 - Asymptotic results for random sums of dependent random variables

AU - Işlak, Ümit

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

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

UR - https://www.scopus.com/pages/publications/84947704369

UR - https://www.scopus.com/pages/publications/84947704369#tab=citedBy

U2 - 10.1016/j.spl.2015.10.015

DO - 10.1016/j.spl.2015.10.015

M3 - Article

AN - SCOPUS:84947704369

SN - 0167-7152

VL - 109

SP - 22

EP - 29

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

ER -

Asymptotic results for random sums of dependent random variables

Abstract

Bibliographical note

Keywords

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