Operator Regular Variation of Multivariate Liouville Distributions

TL;DR

This paper demonstrates that multivariate Liouville distributions exhibit operator regular variation when their driving function is univariate regularly varying, extending the understanding of tail decay phenomena in multivariate distributions.

Contribution

It establishes that multivariate Liouville distributions are operator regularly varying under certain conditions, extending prior results on multivariate regular variation.

Findings

Operator regular variation applies to multivariate Liouville distributions.

The method links density regular variation to distribution regular variation.

Extends the closure property of multivariate regular variation.

Abstract

Operator regular variation reveals general power-law distribution tail decay phenomena using operator scaling, that includes multivariate regular variation with scalar scaling as a special case. In this paper, we show that a multivariate Liouville distribution is operator regularly varying if its driving function is univariate regularly varying. Our method focuses on operator regular variation of multivariate densities, which implies, as we also show in this paper, operator regular variation of the multivariate distributions. This general result extends the general closure property of multivariate regular variation established by de Haan and Resnick in 1987.