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2008 | 28 | 2 | 247-266
Tytuł artykułu

On some limit distributions for geometric random sums

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EN
Abstrakty
EN
We define and give the various characterizations of a new subclass of geometrically infinitely divisible random variables. This subclass, called geometrically semistable, is given as the set of all these random variables which are the limits in distribution of geometric, weighted and shifted random sums. Introduced class is the extension of, considered until now, classes of geometrically stable [5] and geometrically strictly semistable random variables [10]. All the results can be straightforward transfered to the case of random vectors in $ℝ^d$.
Twórcy
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
Bibliografia
  • [1] D. Applebaum, Lévy processes and stochastic calculus, Cambridge Univ. Press, Cambridge 2004.
  • [2] B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for sums of independent random variables, second ed., Addison-Wesley, Reading, Mass.-London 1968.
  • [3] V. Kalashnikov, Geometric sums: Bounds for rare events with applications, Kluwer Academic Publishers, Dordrecht 1997.
  • [4] L.B. Klebanov, G.M. Maniya and I.A. Melamed, A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables, Theory Prob. Appl. 29 (1985), 791-794.
  • [5] T.J. Kozubowski, The inner characterization of geometric stable laws, Statist. Decisions 12 (1994), 307-321.
  • [6] T.J. Kozubowski, Representation and properties of geometric stable laws, Approximation, probability, and related fields, ed. by G. Anastassiou and S.T. Rachev, Plenum Press, New York 1994, pp. 321-337.
  • [7] T.J. Kozubowski and S.T. Rachev, Univariate geometric stable laws, J. Comput. Anal. Appl. 1 (1999), 177-217.
  • [8] G.D. Lin, Characterizations of the Laplace and related distributions via geometric compound, Sankhya Ser. A 56 (1994), 1-9.
  • [9] E. Lukacs, Characteristic functions, second ed., Griffin, London 1970.
  • [10] M.T. Malinowski, Geometrically strictly semistable laws as the limit laws, Discussiones Mathematicae Probability and Statistics 27 (2007), 79-97.
  • [11] M. Maejima and G. Samorodnitsky, Certain probabilistic aspects of semistable laws, Ann. Inst. Statist. Math. 51 (1999), 449-462.
  • [12] D. Mejzler, On a certain class of infinitely divisible distributions, Israel J. Math. 16 (1973), 1-19.
  • [13] N.R. Mohan, R. Vasudeva and H.V. Hebbar, On geometrically infinitely divisible laws and geometric domains of attraction, Sankhyã Ser. A 55 (1993), 171-179.
  • [14] S.T. Rachev and G. Samorodnitsky, Geometric stable distributions in Banach spaces, J. Theoret. Probab. 2 (1994), 351-373.
  • [15] G. Samorodnitsky and M.S. Taqqu, Stable non-gaussian random processes: stochastic models with infinite variance, Chapman and Hall, New York-London 1994.
  • [16] K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Univ. Press, Cambridge 1999.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1103
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