Electronic Journal of Probability

Ratio of the Tail of an Infinitely Divisible Distribution on the Line to that of its Lévy Measure

Toshiro Watanabe and Kouji Yamamuro

Full-text: Open access

Abstract

A necessary and sufficient condition for the tail of an infinitely divisible distribution on the real line to be estimated by the tail of its Lévy measure is found. The lower limit and the upper limit of the ratio of the right tail of an infinitely divisible distribution to the right tail of its Lévy measure are estimated from above and below by reviving Teugels's classical method. The exponential class and the dominated varying class are studied in detail.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 2, 44-74.

Dates
Accepted: 12 January 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819787

Digital Object Identifier
doi:10.1214/EJP.v15-732

Mathematical Reviews number (MathSciNet)
MR2578382

Zentralblatt MATH identifier
1193.60023

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60F99: None of the above, but in this section

Keywords
infinite divisibility L'evy measure $O$-subexponentiality dominated variation exponential class

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Watanabe, Toshiro; Yamamuro, Kouji. Ratio of the Tail of an Infinitely Divisible Distribution on the Line to that of its Lévy Measure. Electron. J. Probab. 15 (2010), paper no. 2, 44--74. doi:10.1214/EJP.v15-732. https://projecteuclid.org/euclid.ejp/1464819787


Export citation

References

  • Albin, J. M. P. A note on the closure of convolution power mixtures (random sums) of exponential distributions. J. Aust. Math. Soc. 84 (2008), no. 1, 1–7.
  • Albin, J. M. P.; Sundén, Mattias. On the asymptotic behaviour of Lévy processes. I. Subexponential and exponential processes. Stochastic Process. Appl. 119 (2009), no. 1, 281–304. (Review)
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation.Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2
  • Braverman, Michael. Suprema and sojourn times of Lévy processes with exponential tails. Stochastic Process. Appl. 68 (1997), no. 2, 265–283.
  • Braverman, Michael. On a class of Lévy processes. Statist. Probab. Lett. 75 (2005), no. 3, 179–189.
  • Braverman, Michael; Samorodnitsky, Gennady. Functionals of infinitely divisible stochastic processes with exponential tails. Stochastic Process. Appl. 56 (1995), no. 2, 207–231.
  • Cline, Daren B. H. Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A 43 (1987), no. 3, 347–365.
  • Cohen, J. W. Some results on regular variation for distributions in queueing and fluctuation theory. J. Appl. Probability 10 (1973), 343–353.
  • Denisov, Denis; Foss, Serguei; Korshunov, Dmitry. On lower limits and equivalences for distribution tails of randomly stopped sums. Bernoulli 14 (2008), no. 2, 391–404.
  • Embrechts, Paul; Goldie, Charles M. On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29 (1980), no. 2, 243–256.
  • Embrechts, Paul; Goldie, Charles M. Comparing the tail of an infinitely divisible distribution with integrals of its Lévy measure. Ann. Probab. 9 (1981), no. 3, 468–481.
  • Embrechts, Paul; Goldie, Charles M. On convolution tails. Stochastic Process. Appl. 13 (1982), no. 3, 263–278.
  • Embrechts, Paul; Goldie, Charles M.; Veraverbeke, Noël. Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49 (1979), no. 3, 335–347.
  • Embrechts, Paul; Hawkes, John. A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory. J. Austral. Math. Soc. Ser. A 32 (1982), no. 3, 412–422.
  • Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas. Modelling extremal events.For insurance and finance.Applications of Mathematics (New York), 33. Springer-Verlag, Berlin, 1997. xvi+645 pp. ISBN: 3-540-60931-8
  • Feller, William. One-sided analogues of Karamata's regular variation. Enseignement Math. (2) 15 1969 107–121.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Foss, Serguei; Korshunov, Dmitry. Lower limits and equivalences for convolution tails. Ann. Probab. 35 (2007), no. 1, 366–383.
  • Goldie, Charles M. Subexponential distributions and dominated-variation tails. J. Appl. Probability 15 (1978), no. 2, 440–442.
  • de Haan, L.; Stadtmüller, U. Dominated variation and related concepts and Tauberian theorems for Laplace transforms. J. Math. Anal. Appl. 108 (1985), no. 2, 344–365.
  • Klüppelberg, Claudia; Kyprianou, Andreas E.; Maller, Ross A. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (2004), no. 4, 1766–1801.
  • Klüppelberg, Claudia; Villaseñor, José A. The full solution of the convolution closure problem for convolution-equivalent distributions. J. Math. Anal. Appl. 160 (1991), no. 1, 79–92.
  • Leslie, J. R. On the nonclosure under convolution of the subexponential family. J. Appl. Probab. 26 (1989), no. 1, 58–66.
  • Pakes, Anthony G. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 (2004), no. 2, 407–424.
  • Rudin, Walter. Limits of ratios of tails of measures. Ann. Probability 1 (1973), 982–994.
  • Sato. K. Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge 1999.
  • Shimura, Takaaki; Watanabe, Toshiro. Infinite divisibility and generalized subexponentiality. Bernoulli 11 (2005), no. 3, 445–469.
  • T. Shimura, T. Watanabe. On the convolution roots in the convolution-equivalent class. The Institute of Statistical Mathematics Cooperative Research Report 175 (2005), pp1-15.
  • Teugels, Jozef L. The class of subexponential distributions. Ann. Probability 3 (1975), no. 6, 1000–1011.
  • Watanabe, Toshiro. Sample function behavior of increasing processes of class $L$. Probab. Theory Related Fields 104 (1996), no. 3, 349–374.
  • Watanabe, Toshiro. Convolution equivalence and distributions of random sums. Probab. Theory Related Fields 142 (2008), no. 3-4, 367–397.
  • Watanabe, Toshiro: Yamamuro, Kouji. Local subexponentiality and self-decomposability. to appear in J. Theoret. Probab. (2010).
  • Yakymiv, A. L. Asymptotic behavior of a class of infinitely divisible distributions.(Russian) Teor. Veroyatnost. i Primenen. 32 (1987), no. 4, 691–702.