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

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

2007-12-04

arxiv.org/abs/0712.0576v2

Annals of Applied Probability 2009, Vol. 19, No. 1, 210-242

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP540 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP540

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.

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