Bulletin of the Korean Mathematical Society (대한수학회보)

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ON THE WEAK LIMIT THEOREMS FOR GEOMETRIC SUMMATIONS OF INDEPENDENT RANDOM VARIABLES TOGETHER WITH CONVERGENCE RATES TO ASYMMETRIC LAPLACE DISTRIBUTIONS

  • Hung, Tran Loc (Department of Mathematics and Statistics Faculty of Economics and Laws University of Finance and Marketing)
  • Received : 2020.11.22
  • Accepted : 2021.05.07
  • Published : 2021.11.30
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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-𝒪" and "small-o" approximation estimates. The obtained results are extensions of some known ones.

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Acknowledgement

The author is grateful to the anonymous referees for carefully reading the manuscript and for offering comments with useful suggestions which allowed him to substantially improve the paper.

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