The Annals of Applied Probability

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

Martin Jacobsen, Thomas Mikosch, Jan Rosiński, and Gennady Samorodnitsky

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In this paper, we consider certain σ-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 σ-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 Lévy motions.

Article information

Ann. Appl. Probab., Volume 19, Number 1 (2009), 210-242.

First available in Project Euclid: 20 February 2009

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Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory
Secondary: 60E07: Infinitely divisible distributions; stable distributions

Cauchy equation Choquet–Deny equation functional equation infinite divisibility infinite moving average inverse problem Lévy measure linear filter regular variation stochastic integral tail of a measure


Jacobsen, Martin; Mikosch, Thomas; Rosiński, Jan; Samorodnitsky, Gennady. Inverse problems for regular variation of linear filters, a cancellation property for σ -finite measures and identification of stable laws. Ann. Appl. Probab. 19 (2009), no. 1, 210--242. doi:10.1214/08-AAP540.

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