On a Multivariate Analog of the Zolotarev Problem

A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented.

Multivariate geometric autoregressive and autoregressive moving average models

We present Autoregressive (AR) and autoregressive moving average (ARMA) processes with multivariate geometric (MG) distribution. The theory of positive dependence is used to show that in many cases, multivariate geometric autoregressive (MGAR) and multivariate autoregressive moving average (MGARMA) models consist of associated random variables. We also provide a special case of the multivariate geometric autoregressive model in which it is stationary and has multivariate geometric distribution.