Multidimensional distributions with heavy tails have recently attracted the attention of several papers on applied probability. About multivariate subexponentiality, we can find several approximations, but none of them have been widely established. Having in mind the single big jump principle, and further the multivariate subexponentiality suggested by [70], we introduce the multivariate long, dominatedly and consistently varying distribution classes. We examine the closure properties of these classes with respect to product convolution, scale mixture and convolution of multivariate distributions. Additionally, for two distributions from the class of multivariate long-tailed distributions, we provide necessary and sufficient conditions in order for their convolution to belong to the class of multivariate subexponential distributions. Furthermore, we study the multivariate, linear, single big jump principle in finite sums and in random sums of random vectors, permitting some dependence structures, which include independence as a special case. Finally, we present an application on the asymptotic behavior of the entrance probability of discounted aggregate claims into some rare sets, in a risk model, with a common Poisson counting process, financial factors and independent, identically distributed claims, with common multivariate subexponential distribution.
