Inverse problems for regular variation of linear filters, a cancellation property for $\sigma$-finite measures and identification of stable laws

TL;DR

This paper investigates how regular variation properties are preserved through linear filters, focusing on cancellation properties of measures and applications to sums, products, and stochastic integrals involving stable laws.

Contribution

It introduces a cancellation property for $\sigma$-finite measures that characterizes when regular variation is preserved under linear transformations.

Findings

Cancellation property is key to regular variation preservation.

Applicable to sums, products, and stochastic integrals.

Provides conditions for the uniqueness of solutions to related functional equations.

Abstract

In this paper, we consider certain $\sigma$-finite measures which can be interpreted as the output of a linear filter. We assume that these measures have regularly varying tails and study whether the input to the linear filter must have regularly varying tails as well. This turns out to be related to the presence of a particular cancellation property in $\sigma$-finite measures, which in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of i.i.d. random variables, to products of independent random variables, and to stochastic integrals with respect to L\'evy motions.