Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-20T12:31:53.675Z Has data issue: false hasContentIssue false
  • English
  • Français

On mixture representation of the Linnik density

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  09 April 2009

Burak Erdoğan  and
I. V. Ostrovskii
I. V. Ostrovskii
Affiliation:
Department of Mathematics Bilkent Universtiy06533 Bilkent AnkaraTurkey e-mail: [email protected] e-mail: [email protected].
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 pα,θ be the Linnik density, that is, the probability density with the characteristic function . The following problem is studied: Let (α θ), (β, ϑ) be two point of PD. When is it possible to represent β,ϑ as a scale mixture of pαθ? A subset of the admissible pairs (α, θ), (β, ϑ) is described.

MSC classification

Secondary: 60E10: Characteristic functions; other transforms 60E05: Distributions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 3 , June 1998 , pp. 317 - 326
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Anderson, D. N. and Arnold, B. C., ‘Linnik distributions and processes’, J. Appl. Probab. 30 (1993), 330–340.CrossRefGoogle Scholar
[2]Arnold, B. C., ‘Some characterizations of the exponential distribution by geometric compounding’, SIAM J. Appl. Math. 24 (1973), 242–244.CrossRefGoogle Scholar
[3]Devroye, L., Non-uniform random variable generation (Springer, New York, 1986).CrossRefGoogle Scholar
[4]Devroye, L., ‘A note on Linnik's distributions’, Statist. Probab. Lett. 9 (1990), 305–306.CrossRefGoogle Scholar
[5]Devroye, L., ‘A triptych of discrete distributions related to the stable law’, Statist. Probab. Lett. 18 (1993), 349–351.CrossRefGoogle Scholar
[6]Erdoğan, B., ‘Analytic and asymptotic properties of non-symmetric Linnik's densities’, (to appear).Google Scholar
[7]Klebanov, L. B., Maniya, G. M. and Melamed, I. A., ‘A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables’, Theory Probab. Appl. 29 (1984), 791–794.CrossRefGoogle Scholar
[8]Kotz, S. and Ostrovskii, I. V., ‘A mixture representation of Linnik's distribution’, Statist. Probab. Lett. 26 (1996), 61–64.CrossRefGoogle Scholar
[9]Kotz, S., Ostrovskii, I. V. and Hayfavi, A., ‘Analytic and asymptotic properties of Linnik's probability density I, II’, J. Math. Anal. Appl. 193 (1995), 353–371, 497–521.CrossRefGoogle Scholar
[10]Kozubowski, T. J., ‘Representation and properties of geometric stable laws,’ in: Approximation, probability, and related fields (Santa Barbara, CA 1993) (eds. Anastassiou, G. and Tachev, S. T.) (Plenum Press, New York, 1994) pp. 321–337.CrossRefGoogle Scholar
[11]Laha, R. G., ‘On a class of unimodal distributions’, Proc. Amer. Math. Soc. 12 (1961), 181–184.CrossRefGoogle Scholar
[12]Linnik, Ju. V., ‘Linear forms and statistical criteria, II’, Select. Transl. Math. Stat. Probab. 3 (1963), 41–90 (Original paper appeared in: Ukrainskii Mat. Zhournal, 5 (1953), 247–290.)Google Scholar
[13]Linnik, Ju. V. and Ostrovskii, I. V., Decomposition of random variables and vectors (Amer. Math. Soc., Providence, 1977).Google Scholar
[14]Oberhettinger, F., Tables of Fourier transforms and Fouier transforms of distributions (Springer, Berlin, 1990).CrossRefGoogle Scholar
[15]Pakes, A. G., ‘A characterization of gamma mixtures of stable laws motivated by limit theorems’, Statist. Neerlandica 2–3 (1992), 209–218.CrossRefGoogle Scholar
[16]Pakes, A. G., ‘On characterizations via mixed sums’, Austral. J. Statist. 34 (1992), 323–339.CrossRefGoogle Scholar
[17]Pakes, A. G., ‘Necessary conditions for characterization of laws via mixed sums’, Ann. Inst. Statist. Math. 46 (1994), 797–802.CrossRefGoogle Scholar
[18]Pakes, A. G., ‘Characterization of discrete laws via mixed sums and Markiv branching process’, Stochastic Process. Appl. 55 (1995), 285–300.CrossRefGoogle Scholar
[19]Zolotarev, V. M., One-dimensional stable distributions (Amer. Math. Soc., Providence, 1986).CrossRefGoogle Scholar