Exact Asymptotics of Bivariate Scale Mixture Distributions

TL;DR

This paper derives precise asymptotic formulas for the tail probabilities of bivariate scale mixture distributions, especially when the scaling variable follows a Gumbel max-domain distribution, with applications to polar vectors and dependence analysis.

Contribution

It provides the first exact asymptotic expansions for tail probabilities of bivariate scale mixtures with R in the Gumbel domain, including special cases like polar vectors.

Findings

Exact asymptotic tail behavior for bivariate scale mixtures derived.

Results applied to analyze asymptotic independence and conditional excess.

Special case retrieval for bivariate polar random vectors.

Abstract

Let (RU_1, R U_2) be a given bivariate scale mixture random vector, with R>0 being independent of the bivariate random vector (U_1,U_2). In this paper we derive exact asymptotic expansions of the tail probability P{RU_1> x, RU_2> ax}, a \in (0,1] as x tends infintiy assuming that R has distribution function in the Gumbel max-domain of attraction and (U_1,U_2) has a specific tail behaviour around some absorbing point. As a special case of our results we retrieve the exact asymptotic behaviour of bivariate polar random vectors. We apply our results to investigate the asymptotic independence and the asymptotic behaviour of conditional excess for bivariate scale mixture distributions.