Aggregating heavy-tailed random vectors: from finite sums to L\'evy processes

TL;DR

This paper extends the understanding of tail behavior in heavy-tailed random vectors and L\'evy processes, showing that aggregate tail probabilities can be influenced by multiple large jumps, not just one.

Contribution

It generalizes the principle of 'one large jump' to a broader context involving multivariate regular variation on subcones, revealing more complex tail behaviors.

Findings

Aggregates of heavy-tailed vectors are multivariate regularly varying on subcones.

Tail probabilities can be approximated even when classical regular variation suggests negligibility.

Multiple large jumps can influence the tail behavior depending on the event structure.

Abstract

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, L\'evy 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 quadrant, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities which were rendered asymptotically negligible under classical regular variation, despite the "one large jump'' asymptotics. We also discover that depending on the structure of the tail event of concern, the tail behavior of the aggregates may be characterized by more than a single large jump. Eventually, we illustrate a similar phenomenon for multivariate regularly varying L\'evy processes, establishing as well a relationship between multivariate regular variation of a L\'evy process and multivariate regular variation of its L\'evy measure on different subcones.