On the weak limit theorems for geometric summations of independent random variables together with convergence rates to asymmetric Laplace distributions. (English) Zbl 1492.60124
Summary: 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.
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60F05 | Central limit and other weak theorems |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 41A36 | Approximation by positive operators |
Keywords:
geometric summation; asymmetric Laplace distribution; weak limit theorem; geometric Lindeberg condition; rate of convergence; Trotter’s operatorReferences:
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