ИСТИНА

This distribution has been discovered by I.N. Kovalenko[1] in 1965. Using Laplaсe transforms he has described all possible limit laws for the distributions of appropriately normalized geometric sums of independent identically distributed nonnegative random variables. Twenty five years later this limit property of Mittag-Leffler distributions was re-discovered by A. Pillai[2] who propose the term Mittag-Leffler distribution for this type of distributions. Recently Korolev and Zeifman[3] have found many interesting relations of Mittag-Leffler distributions with stable, Laplace, Linnik and Weibull distributions. It is not difficult to get the analogous results for the random sums of random vectors when the summation index has univariate geometric distribution. In our report we consider more interesting situation of multivariate random sums where summation index has multivariate geometric distribution. Using this approach we get new and more general variants of multivariate Mittag-Leffler, Laplace, Linnik and Weibull distributions. This research is supported by RFCF, project 18-11-00155 Keywords: Multivariate random sums, multivariate geometric distribution, scale mixtures, subordinated processes. References 1. Kovalenko I.N. On the class of limit distributions for rarefied flows of homogeneous events. Litovskii Mathematicheskii Sbornik (Lithuanian Mathematical Journal) , 5, 4, 569-573, 1965. 2. Pillai R.N. On Mittag-Leffler functions and related distributions. Annals of the Institute of Statistical Mathematics, 42, 157-161, 1990. 3. Korolev V. Yu. and Zeifman A.I. A Note on Mixture Representations for the Linnik and Mittag-Leffler Distributions and Their Applications. Journal of Math. Sciences. 218, 314-327, 2016.