Gaussian copula of stable random vectors and application

Abstract

In this paper, we present a new method to investigate data of multivariate heavy-tailed distributions. We show that for any given number $\alpha \in (0;2]$, each Gaussian copula is also the copula of an $\alpha$-stable random vector. Simultaneously, every random vector is $\alpha$-stable if its marginals are $\alpha$-stable and its copula is a Gaussian copula. The result is used to build up a formula representing density functions of $\alpha$-stable random vectors with Gaussian copula. Adopting a new tool, the paper points out that pairs of GPS signals recording latitude and longitude of a fixed point have two-dimensional stable distribution, and in the most of cases, vectors of daily returns in stock market data have multivariate stable distributions with Gaussian copulas.