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Multivariate subexponential distributions and random sums of random vectors

Abstract:

Let F(x) denote a distribution function in ℝd and let F*n(x) denote the nth convolution power of F(x). In this paper we discuss the asymptotic behaviour of 1 - F* n(x) as x tends to ∞ in a certain prescribed way. It turns out that in many cases 1 - F*n(x) ∼ n(1 - F(x)). To obtain results of this type, we introduce and use a form of subexponential behaviour, thereby extending the notion of multivariate regular variation. We also discuss subordination, in which situation the index n is replaced by a random index N. © Applied Probability Trust 2006.