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V.Yu. Korolev,A.I. Zeifman 한국통계학회 2017 Journal of the Korean Statistical Society Vol.46 No.2
We present some mixture representations for the Linnik, Mittag-Leffler and Weibull distributions in terms of normal, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e.g., random sums of independent random variables with finite variances, maximum random sums, extreme order statistics, in which the Linnik and Mittag-Leffler distributions play the role of limit laws. Thus we demonstrate that the scheme of geometric summation is far not the only asymptotic setting (even for sums of independent random variables) in which the Mittag-Leffler and Linnik laws appear as limit distributions. The two-sided Mittag-Leffler and the one-sided Linnik distribution are introduced and also proved to be limit laws for some statistics constructed from samples with random sizes.
On asymmetric generalization of the Weibull distribution by scale–location mixing of normal laws
V.Yu. Korolev,Lily Kurmangazieva,A.I. Zeifman 한국통계학회 2016 Journal of the Korean Statistical Society Vol.45 No.2
Two approaches are suggested to the definition of asymmetric generalized Weibull distribution. These approaches are based on the representation of the two-sided Weibull distributions as variance–mean normal mixtures or more general scale–location mixtures of the normal laws. Since both of these mixtures can be limit laws in limit theorems for random sums of independent random variables, these approaches can provide additional arguments in favor of asymmetric two-sided Weibull-type models of statistical regularities observed in some problems related to stopped random walks, in particular, in problems of modeling the evolution of financial markets