Multivariate subexponential distributions

We present a formulation of subexponential and exponential tail behavior for multivariate distributions. The definitions are necessarily in terms of vague convergence of Radon measures rather than of ratios of distribution tails. With the proper setting, we show that if all one dimensional marginals of a d-dimensional distribution are subexponential, then the distribution is multivariate subexponential. Known results for univariate subexponential distributions are extended to the multivariate setting. Point process arguments are used for the proofs.