On the weak limit theorems for geometric summations of independent random variables together with convergence rates to asymmetric Laplace distributions
Bull. Korean Math. Soc.
Published online July 13, 2021
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 order of approximations of the distributions of geometric summations by the asymmetric Laplace distributions are established in term of the "large-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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