Domain of Attraction of -stable Distributions under Finite Mixture Models

C.E.G. Otiniano, C.R. Gonçalves

Abstract


In this work, we study the asymptotic distribution of the normalized sum of independent, identically distributed random variables under the finite mixture models. In the Theorem we give necessary conditions for a distribution function of a mixed population with k components to belong to the domain of attraction of an α-stable distribution, by assuming that each component of the mixture also pertains to the domain of attraction of an α-stable distribution. Examples are given to illustrate the result.

References


P. Embrechts, T. Mikosch, C. Kluppelberg, “Modelling Extremal Events for Insurance and Finance”, Springer - Verlag, Berlin, 1997.

B.S. Everitt, D.J. Hand, “Finite Mixture Distributions”, Chapman and Hall, London, 1981.

W. Feller, “An Introduction to Probability Theory and its Applications II”, 3rd edition, Wiley, New York, 1968.

I.A. Ibragimov, Yu. V. Linnik, “Independent and Stationary Sequences of Ran- dom Variables”, Wolters-Noordhoff, Groningen, 74–85, 1971.

B. G. Lindsay, “Mixture Models: Theory, Geometry and Applications”, Insti- tute of Mathematical and Statistics, Hayward, CA, 1995.

G.J. McLachlan, K.E.Basford, “Mixture Models: Aplications to Clustering”, Marcel Dekker, New York, 1988.

N. Ravishanker, D.K. Dey, Multivariate survival models with a mixture of positive stable frailties, Journal Methodology and Computing in Applied Probability, 2 (2004), 293–308.

G. Samorodnitsky, M. Taqqu, “Stable Non-Gaussian Random Process”, Chap- man and Hall, New York, 1994.

D.M. Titterington, A.F.M. Smith, U.E. Makov, “Statistical Analysis of Finite Mixture Distributions”, Wiley, New York, 1985.

V. M. Zolotarev, “One Dimensional Stable Distributions”, American Mathematical Society, Providence, 1986.




DOI: https://doi.org/10.5540/tema.2010.011.01.0069

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Trends in Computational and Applied Mathematics

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