Multivariate strong subexponential distributions: properties and applications

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.