Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(6): 1419-1443

Published online November 30, 2021 https://doi.org/10.4134/BKMS.b200980

Copyright © The Korean Mathematical Society.

On the weak limit theorems for geometric summations of independent random variables together with convergence rates to asymmetric Laplace distributions

Tran Loc Hung

University of Finance and Marketing

Abstract

The asymmetric Laplace distribution arises as a limiting distribution of geometric summations of independent and identically distributed random variables with finite second moments. The main purpose of this paper is to study the weak limit theorems for geometric summations of independent (not necessarily identically distributed) random variables together with convergence rates to asymmetric Laplace distributions. Using Trotter-operator method, the orders of approximations of the distributions of geometric summations by the asymmetric Laplace distributions are established in term of the ``large--$\mathcal{O}$" and ``small--o" approximation estimates. The obtained results are extensions of some known ones.

Keywords: Geometric summation, asymmetric Laplace distribution, weak limit theorem, geometric Lindeberg condition, rate of convergence, Trotter's operator

MSC numbers: Primary 60G50, 60F05, 60E07; Secondary 41A36

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