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Functionals of infinitely divisible stochastic processes with exponential tails

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  • Braverman, Michael
  • Samorodnitsky, Gennady

Abstract

We investigate the tail behavior of the distributions of subadditive functionals of the sample paths of infinitely divisible stochastic processes when the Lévy measure of the process has suitably defined exponentially decreasing tails. It is shown that the probability tails of such functionals are of the same order of magnitude as the tails of the same functionals with respect to the Lévy measure, and it turns out that the results of this kind cannot, in general, be improved. In certain situations we can further obtain both lower and upper bounds on the asymptotic ratio of the two tails. In the second part of the paper we consider the particular case of Lévy processes with exponentially decaying Lévy measures. Here we show that the tail of the maximum of the process is, up to a multiplicative constant, asymptotic to the tail of the Lévy measure. Most of the previously published work in the area considered heavier than exponential probability tails.

Suggested Citation

  • Braverman, Michael & Samorodnitsky, Gennady, 1995. "Functionals of infinitely divisible stochastic processes with exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 207-231, April.
    • Handle: RePEc:eee:spapps:v:56:y:1995:i:2:p:207-231
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    References listed on IDEAS

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    1. Berman, Simeon M., 1986. "The supremum of a process with stationary independent and symmetric increments," Stochastic Processes and their Applications, Elsevier, vol. 23(2), pages 281-290, December.
    2. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    3. Willekens, Eric, 1987. "On the supremum of an infinitely divisible process," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 173-175.
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    Cited by:

    1. Braverman, Michael, 2000. "Suprema of compound Poisson processes with light tails," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 145-156, November.
    2. Braverman, Michael, 1999. "Remarks on suprema of Lévy processes with light tailes," Statistics & Probability Letters, Elsevier, vol. 43(1), pages 41-48, May.
    3. Albin, J. M. P., 1999. "Extremes of totally skewed [alpha]-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 185-212, February.
    4. Braverman, Michael, 1997. "Suprema and sojourn times of Lévy processes with exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 265-283, June.
    5. Braverman, Michael, 2005. "On a class of Lévy processes," Statistics & Probability Letters, Elsevier, vol. 75(3), pages 179-189, December.
    6. Fasen, Vicky, 2006. "Extremes of subexponential Lévy driven moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 116(7), pages 1066-1087, July.
    7. Braverman, Michael, 2010. "On suprema of Lévy processes with light tails," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 541-573, April.
    8. Embrechts, Paul & Samorodnitsky, Gennady, 1995. "Sample quantiles of heavy tailed stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 217-233, October.
    9. Albin, J.M.P. & Sundén, Mattias, 2009. "On the asymptotic behaviour of Lévy processes, Part I: Subexponential and exponential processes," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 281-304, January.
    10. Griffin, Philip S. & Maller, Ross A. & Roberts, Dale, 2013. "Finite time ruin probabilities for tempered stable insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 478-489.

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