Systems and Means of Informatics

2026, Volume 36, Issue 2, pp 84-105

MULTIVARIATE QUASI-EXPONENTIATED MIXED NORMAL DISTRIBUTIONS

M. A. Ivanov
V. Yu. Korolev

Abstract

A class of multivariate elliptically contoured distributions is introduced and studied. Each one-dimensional projection of such a distribution has the quasi-exponentiated normal distribution that coincides with the distribution of the radom variable

Q is the positive random variable; and X is the random variable with the standard normal distribution independent of Q. For

> 1, the densities of the multivariate mixed normal distributions are infinite in zero. This property makes it possible to use multivariate quasi-exponentiated mixed power distributions with

> 1 as models of statistical regularities in the behavior of multivariate stochastic processes with rather long periods within which the process either does not change or changes insignificantly, alternate with the periods when variations with rather large jumps are observed. Unlike "pure" quasi-exponentiated distributions, quasi-exponentiated mixed normal distributions possess heavy tails that may be useful in the case of very large jumps of the process under consideration. Some limit theorems are presented on convergence of the distributions of multivariate statistics constructed from samples with random sizes, including random sums, to multivariate quasi-exponentiated mixed normal distributions. As an example, multivariate elliptically contoured quasi-exponentiated logistic distribution is considered.

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