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2007 | 27 | 1-2 | 79-97
Tytuł artykułu

Geometrically strictly semistable laws as the limit laws

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EN
Abstrakty
EN
A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable $X_p$ such that $X\stackrel{d}{=} ∑_{k=1}^{T(p)}X_{p,k}$, where $X_{p,k}$'s are i.i.d. copies of $X_p$, and random variable T(p) independent of ${X_{p,1},X_{p,2},...}$ has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The main results concern geometrically strictly semistable distributions which form a subset of geometrically infinitely divisible distributions. We show that they are limit laws for random and deterministic sums of independent random variables.
Twórcy
  • Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Szafrana 4a, 65-516 Zielona Góra, Poland
Bibliografia
  • [1] I. Gertsbach, A survey of methods used in engineering reliability, Proceedings 'Reliability and Decision Making', Universita di Siena 1990.
  • [2] L.B. Klebanov, G.M. Maniya and I.A. Melamed, A problem of V.M. Zolotarev and analogues of infinitely divisible and stable distributions in a scheme for summation of a random number of random variables, Theory Probab. Appl. 29 (1984), 791-794.
  • [3] P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris 1937.
  • [4] G.D. Lin, Characterizations of the Laplace and related distributions via geometric compound, Sankhya Ser. A 56 (1994), 1-9.
  • [5] M. Loève, Nouvelles classes de lois limites, Bull. Soc. Math. France 73 (1945), 107-126.
  • [6] M. Maejima, Semistable distributions, in: Lévy Processes, Birkhäuser, Boston 2001, 169-183.
  • [7] M. Maejima and G. Samorodnitsky, Certain probabilistic aspects of semistable laws, Ann. Inst. Statist. Math. 51 (1999), 449-462.
  • [8] N.R. Mohan, R. Vasudeva and H.V. Hebbar, On geometrically infinitely divisible laws and geometric domains of attraction, Sankhya Ser. A 55 (1993), 171-179.
  • [9] S.T. Rachev and G. Samorodnitsky, Geometric stable distributions in Banach spaces, J. Theoret. Probab. 2 (1994), 351-373.
  • [10] K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Univ. Press, Cambridge 1999.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1089
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