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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Abstract

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

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

Dates
First available in Project Euclid: 20 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1235140338

Digital Object Identifier
doi:10.1214/08-AAP540

Mathematical Reviews number (MathSciNet)
MR2498677

Zentralblatt MATH identifier
1171.60309

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

Keywords
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

Citation

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. https://projecteuclid.org/euclid.aoap/1235140338


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References

  • [1] Barndorff-Nielsen, O. E., Rosiński, J. and Thorbjørnsen, S. (2008). General ϒ-transformations. ALEA Lat. Am. J. Probab. Math. Stat. 4 131–165.
  • [2] Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Probab. 12 908–920.
  • [3] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [4] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and its Applications 27. Cambridge Univ. Press, Cambridge.
  • [5] Bingham, N. H. and Inoue, A. (2000). Tauberian and Mercerian theorems for systems of kernels. J. Math. Anal. Appl. 252 177–197.
  • [6] Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323–331.
  • [7] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • [8] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179–195.
  • [9] Davis, R. and Resnick, S. (1985). More limit theory for the sample correlation function of moving averages. Stochastic Process. Appl. 20 257–279.
  • [10] Davis, R. and Resnick, S. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14 533–558.
  • [11] Denisov, D. and Zwart, B. (2005). On a theorem of Breiman and a class of random difference equations. Technical report.
  • [12] Doob, J. L. (1942). The Brownian movement and stochastic equations. Ann. of Math. (2) 43 351–369.
  • [13] Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29 243–256.
  • [14] Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49 335–347.
  • [15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [16] Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38 1466–1474.
  • [17] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. II, 2nd ed. Wiley, New York.
  • [18] Jessen, A. H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80(94) 171–192.
  • [19] Lukacs, E. (1970). Characteristic Functions, 2nd ed, revised and enlarged. Hafner Publishing Co., New York.
  • [20] Maulik, K. and Resnick, S. (2004). Characterizations and examples of hidden regular variation. Extremes 7 31–67 (2005).
  • [21] Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10 1025–1064.
  • [22] Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford Studies in Probability 4. Oxford Univ. Press, New York.
  • [23] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451–487.
  • [24] Rao, C. R. and Shanbhag, D. N. (1994). Choquet-Deny Type Functional Equations with Applications to Stochastic Models. Wiley, Chichester.
  • [25] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • [26] Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996–1014.
  • [27] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Stochastic Modeling. Chapman and Hall, New York.
  • [28] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. Translated from the 1990 Japanese original, revised by the author.
  • [29] Shimura, T. (2000). The product of independent random variables with regularly varying tails. Acta Appl. Math. 63 411–432.
  • [30] Steutel, F. W. (1979). Infinite divisibility in theory and practice. Scand. J. Statist. 6 57–64.
  • [31] Willekens, E. (1987). On the supremum of an infinitely divisible process. Stochastic Process. Appl. 26 173–175.
  • [32] Zemanian, A. H. (1987). Distribution Theory and Transform Analysis, 2nd ed. Dover, New York.