Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T15:48:48.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular variation of the tail of a subordinated probability distribution

Published online by Cambridge University Press:  01 July 2016

A. J. Stam
A. J. Stam*
Affiliation:
State University of Groningen
Get access Rights & Permissions [Opens in a new window]

Abstract

Let F be a probability measure on the real line and G = Σ C(k)Fk∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V(x) vary regularly of order p for x→ ∞ and either (1) V(x) F[x, ∞)→ α ≧ 0 or (2) V(x) C[x, ∞)→ γ ≧ 0. If ρ > 1 and F(–∞, 0) = 0, necessary and sufficient in order that V(x) G[x, ∞)→b, is that both (1) and (2) hold for suitable α and γ. For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.

Keywords

SUBORDINATIONREGULAR VARIATIONCONVOLUTIONTAIL BEHAVIOURRENEWAL THEORYSTABLE LAWSATTRACTION
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 2 , August 1973 , pp. 308 - 327
Copyright
Copyright © Applied Probability Trust 1973 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Balkema, A. A. and De Haan, L. (1972) Residual life time at great age. Technical Report no. 36, Dept. of Statistics, Stanford University.Google Scholar
[2] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass. Google Scholar
[3] Callaert, H. and Cohen, J. W. (1972) A lemma on regular variation of a transient renewal function. Zeit. Wahrscheinlichkeitsth. To appear.CrossRefGoogle Scholar
[4] Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[5] Cohen, J. W. (1972) On the tail of the stationary waiting time distribution and limit theorems for the M/G/I queue. Ann. Inst. H. Poincaré B 8, 3. To appear.Google Scholar
[6] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II, 1st. ed. Wiley, New York.Google Scholar
[7] De Haan, L. (1970) On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts no. 32, Mathematisch Centrum, Amsterdam.Google Scholar
[8] De Haan, L. (1971) A form of regular variation and its application to the domain of attraction of the double exponential distribution. Zeit. Wahrscheinlichkeitsth. 17, 241–258.CrossRefGoogle Scholar
[9] De Haan, L. (1971) Weak limits of residual life times. Report SW 7/71, Mathematisch Centrum, Amsterdam.Google Scholar
[10] Ibragimov, I. A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
[11] Kalma, J. N. (1972) Generalized Renewal Measures. Groningen University.Google Scholar
[12] Loève, M. (1963) Probability Theory. 3rd ed. Van Nostrand, Princeton.Google Scholar
[13] Nevels, K. (1971) On attraction by stable laws I. Report TW–104, Mathematisch Instituut, Rijksuniversiteit, Groningen.Google Scholar
[14] Stam, A. J. (1965) Derived stochastic processes. Comp. Math. 17, 102–140.Google Scholar
[15] Stam, A. J. (1972) The tail of the ladder height distribution. Report TW–107, Mathematisch Instituut, Rijksuniversiteit, Groningen.Google Scholar
[16] Teugels, J. L. (1969) Abel-Tauber methoden en subordinatie van stochastische processes Katholieke Universiteit, Leuven.Google Scholar
[17] Teugels, J. L. (1970) Regular variation of Markov renewal functions. J. London Math. Soc. 2, 179–190.CrossRefGoogle Scholar