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An Extension of Zolotarev’s Problem and Some Related Results

Authors
  • Hung, Tran Loc1
  • Kien, Phan Tri1
  • 1 University of Finance and Marketing, 77 Nguyen Kiem Street, Phu Nhuan District, Ho Chi Minh City, Vietnam , Ho Chi Minh City (Vietnam)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 29, 2021
Volume
41
Issue
5
Pages
1619–1634
Identifiers
DOI: 10.1007/s10473-021-0513-6
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

The main purpose of this paper is to extend the Zolotarev’s problem concerning with geometric random sums to negative binomial random sums of independent identically distributed random variables. This extension is equivalent to describing all negative binomial infinitely divisible random variables and related results. Using Trotter-operator technique together with Zolotarev-distance’s ideality, some upper bounds of convergence rates of normalized negative binomial random sums (in the sense of convergence in distribution) to Gamma, generalized Laplace and generalized Linnik random variables are established. The obtained results are extension and generalization of several known results related to geometric random sums.

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