Multivariate strong subexponential distributions: properties and applications
Charalampos D. Passalidis
TL;DR
The paper develops the theory of multivariate strong subexponential distributions on fixed sets A and establishes their core properties, including a multivariate Kesten-type inequality and convolution-closure relations. It then proves a multivariate single-big-jump principle for randomly stopped sums under weaker tail conditions than in the standard subexponential setting, and provides uniform asymptotics for precise large deviations in both nonrandom and random multivariate sums. An application to a nonstandard multivariate risk model with constant interest illustrates the practical impact, delivering uniform ruin- and entrance-probability approximations for rare events. Collectively, the results extend univariate heavy-tailed theory to multivariate contexts and offer robust tools for multivariate risk assessment under dependent heavy tails.
Abstract
In this paper we introduce and study the class of multivariate strong subexponential distributions and of multivariate strongly subexponential distributions. Some first properties are verified, as for example a type of multivariate analogue of Kesten's inequality, the closure property with respect to convolution, and the conditional closure property with respect to convolution roots. Next, we establish the the single big jump principle for the randomly stopped sums, under the assumption that the random vectors in the summation belong to the class of multivariate strong subexponential distributions. Here the conditions of the counting random variable are weaker in comparison with them in multivariate subexponential class. Further, we establish uniform asymptotic estimates for the precise large deviations in multivariate set up, both for random and nonrandom sums, when the distribution of the summands belongs to the class of multivariate strongly subexponential distributions. Finally, we provide an application in a nonstandard risk model, with independent and identically distributed claim vectors, from the class of multivariate strong subexponential distributions and in the presence of constant interest force.