×

zbMATH — the first resource for mathematics

Convolution equivalence and infinite divisibility. (English) Zbl 1051.60019
Summary: Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

MSC:
60E07 Infinitely divisible distributions; stable distributions
60F99 Limit theorems in probability theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation . Cambridge University Press. · Zbl 0617.26001
[2] Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions. Springer, New York. · Zbl 0756.60015
[3] Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10 , 323–331. · Zbl 0147.37004
[4] Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Prob . Appl. 9 , 640–648. · Zbl 0203.19401
[5] Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Prob. 1 , 663–673. · Zbl 0387.60097
[6] Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob . Theory Relat. Fields 72 , 529–557. · Zbl 0577.60019
[7] Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A 43 , 347–365. · Zbl 0633.60021
[8] 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. · Zbl 0425.60011
[9] Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13 , 263–278. · Zbl 0487.60016
[10] Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49 , 335–347. · Zbl 0397.60024
[11] Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions . In A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions , eds R. Adler, R. Feldman and M. S. Taqqu, Birkhäuser, Boston, MA, pp. 435–460. · Zbl 0923.62021
[12] Mantegna, R. N. and Stanley, H. E. (1994). Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. Phys. Rev. Lett. 73 , 2946–2949. · Zbl 1020.82610
[13] Pakes, A. G. (2003). Investigating the structure of truncated Lévy-stable laws. In Science and Statistics: A Festschrift for Terry Speed (Lecture Notes Monogr. Ser. 40 ), ed. D. R. Goldstein, Institute of Mathematical Statistics, Beachwood, OH, pp. 49–78.
[14] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester. · Zbl 0940.60005
[15] Sato, K. (1999). L évy Processes and Infinitely Divisible Distributions. Cambridge University Press. · Zbl 0973.60001
[16] Sigman, K. (1999). Editorial introduction (Special issue on queues with heavy-tailed distributions). Queueing Systems 33 , 1–3. · Zbl 0997.60118
[17] Stam, A. J. (1973). Regular variation of the tail of a subordinated distribution. Adv. Appl. Prob. 5 , 308–327. · Zbl 0264.60029
[18] Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3 , 1000–1011. · Zbl 0374.60022
[19] Willekens, E. (1987). Subexponentiality on the real line. Res. Rep., Katholieke Universiteit Leuven. · Zbl 0633.60025
[20] Yakymiv, A. L. (2002). On the asymptotics of the density of an infinitely divisible distribution at infinity. Theory Prob. Appl. 47 , 114–122. · Zbl 1033.60010
[21] Cai, J. and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41 , 117–130. · Zbl 1054.60012
[22] Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13 , 37–53. · Zbl 1045.60039
[23] Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25 , 132–141. · Zbl 0651.60020
[24] Sgibnev, M. S. (1990). Asymptotics of infinitely divisible distributions in \(\mathbf R\). Siberian Math. J. 31 , 115–119. · Zbl 0714.60010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.