Aggregating heavy-tailed random vectors: from finite sums to lévy processes

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of ‘one large jump’, be it for finite sums, random sums, or Lévy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are multivariate regularly varying on various subcones of the positive orthant [0,∞)$^𝑑$, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities rendered asymptotically negligible under classical regular variation. Second, we discover that depending on the structure of a particular tail event, the tail behavior of the aggregates may be characterized by more than a single large jump. Finally, we illustrate a similar phenomenon for regularly varying multivariate Lévy processes, establishing as well a relationship between regular variation of a multivariate Lévy process and multivariate regular variation of its Lévy measure on different subcones. The applicability of these results in financial and insurance risk management is discussed.