Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-13T13:38:48.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate subexponential distributions and random sums of random vectors

Part of:Real functionsDistribution theory - ProbabilitySpecial processes

Published online by Cambridge University Press:  08 September 2016

E. Omey,
F. Mallor and
J. Santos
E. Omey*
Affiliation:
European University College Brussels
F. Mallor*
Affiliation:
Public University of Navarre
J. Santos*
Affiliation:
Public University of Navarre
*
∗ Postal address: Department of Mathematics and Statistics, European University College Brussels, Stormstraat 2, 1000 Brussels, Belgium.
∗∗ Postal address: Department of Statistics and Operations Research, Public University of Navarre, Campus Arrosadia, 31006 Pamplona, Spain. Email address: mallor@unavarra.es
∗∗∗ Current address: Los Alderetes 14, 4° 3, 14004 Cordoba, Spain.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let F(x ) denote a distribution function in R d and let F *n (x ) denote the nth convolution power of F(x ). In this paper we discuss the asymptotic behaviour of 1 - F *n (x ) as x tends to ∞ in a certain prescribed way. It turns out that in many cases 1 - F *n (x ) ∼ n(1 - F(x )). To obtain results of this type, we introduce and use a form of subexponential behaviour, thereby extending the notion of multivariate regular variation. We also discuss subordination, in which situation the index n is replaced by a random index N.

Keywords

Subexponential distributionregular variationrandom sum

MSC classification

Primary:26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Secondary:60E99: None of the above, but in this section60K99: None of the above, but in this section

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006, pp. 1028 - 1046
Copyright
Copyright © Applied Probability Trust 2006 

References

Alpár, L. (1984). Tauberian theorems for power series of two variables. Studia Sci. Math. Hung. 19, 165–176.Google Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908–920.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.CrossRefGoogle Scholar
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.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Anal. Math. 26, 255–302.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.Google Scholar
Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. A 43, 347–365. (Correction: 48 (1990), 152–153.)CrossRefGoogle Scholar
Cline, D. B. H. and Resnick, S. I. (1992). Multivariate subexponential distributions. Stoch. Process. Appl. 42, 49–72.CrossRefGoogle Scholar
De Haan, L. and Omey, E. (1984). Integrals and derivatives of regularly varying functions in R d and domains of attraction of stable distributions. II. Stoch. Process. Appl. 16, 157–170.CrossRefGoogle Scholar
De Haan, L., Omey, E. and Resnick, S. I. (1984). Domains of attraction and regular variation in R d . J. Multivariate Anal. 14, 17–33.Google Scholar
Embrechts, P. (1985). Subexponential distribution functions and their applications: a review. In Proc. 7th Conf. Prob. Theory (Brasov, Romania, 1982), eds Iosifescu, M. et al., VSP, Zeist, pp. 125–136.Google Scholar
Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. A 29, 243–256.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13, 263–278.Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, New York.CrossRefGoogle Scholar
Geluk, J. L. and De Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems (CWI Tracts 40). Centre For Mathematics and Computer Science, Amsterdam.Google Scholar
Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails, eds Adler, R. J., Feldman, R. E. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 435–459.Google Scholar
Hult, H. and Linskog, F. (2006). On Kesten's counterexample to the Cramér–Wold device for regular variation. Bernoulli 12, 133–142.Google Scholar
Johns, M. V. Jr. (1957). Non-parametric empirical Bayes procedures. Ann. Math. Statist. 28, 649–669.Google Scholar
Mallor, F. and Omey, E. (2001). Shocks, runs and random sums. J. Appl. Prob. 38, 438–448.Google Scholar
Mallor, F. and Omey, E. (2002). Shocks, runs and random sums: asymptotic behaviour of the distribution function. J. Math. Sci. NY 111, 3559–3565.Google Scholar
Mallor, F., Omey, E. and Santos, J. (2006). Asymptotic results for a run and cumulative mixed shock model. J. Math. Sci. 138, 5410–5415.Google Scholar
Mallor, F., Omey, E. and Santos, J. (2007). Multivariate weighted renewal functions. J. Multivariate Anal. 98, 30–39.Google Scholar
Omey, E. (1982). Multivariate Reguliere Variatie en Toepassingen in Kanstheorie. Doctoral Thesis. Katholieke Universiteit Leuven.Google Scholar
Omey, E. (1989). Multivariate Regular Variation and Applications in Probability Theory (Eclectica 74). EHSAL, Brussels.Google Scholar
Omey, E. (1990). Random sums of random vectors. Publ. Inst. Math. 48, 191–198.Google Scholar
Omey, E. (2006). Subexponential distributions and the difference between the product and the convolution product of distribution functions in R d . J. Math. Sci. 138, 5434–5449.CrossRefGoogle Scholar
Omey, E. and Teugels, J. L. (2002). Weighted renewal functions: a hierarchical approach. Adv. Appl. Prob. 34, 394–415.CrossRefGoogle Scholar
Omey, E. and Willekens, E. (1989). Abelian and Tauberian theorems for the Laplace transform of functions in several variables. J. Multivariate Anal. 30, 292–306.CrossRefGoogle Scholar
Pratt, J. W. (1960). On interchanging limits and integrals. Ann. Math. Statist. 31, 74–77.Google Scholar
Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes (Appl. Prob. 4). Springer, New York.Google Scholar
Santos, J. (2004). Fiabilidad de sistemas sometidos a choques. Doctoral Thesis, Public University of Navarre, Pamplona.Google Scholar
Seneta, E. (1976). Regularly Varying Functions (Lecture Notes Math. 508). Springer, New York.CrossRefGoogle Scholar
Stam, A. J. (1977). Regular variation in R d and the Abel–Tauber theorem. Tech. Rep. TW 189, Mathematics Institute, University of Groningen.Google Scholar
Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3, 1000–1011.Google Scholar