Geometrically strictly semistable laws as the limit laws

Marek T. Malinowski

Discussiones Mathematicae Probability and Statistics (2007)

  • Volume: 27, Issue: 1-2, page 79-97
  • ISSN: 1509-9423

Abstract

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A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable X p such that X = 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.

How to cite

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Marek T. Malinowski. "Geometrically strictly semistable laws as the limit laws." Discussiones Mathematicae Probability and Statistics 27.1-2 (2007): 79-97. <http://eudml.org/doc/277030>.

@article{MarekT2007,
abstract = {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.},
author = {Marek T. Malinowski},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {infinite divisibility; geometric infinite divisibility; geometric semistability; random sums; limit laws; characteristic function},
language = {eng},
number = {1-2},
pages = {79-97},
title = {Geometrically strictly semistable laws as the limit laws},
url = {http://eudml.org/doc/277030},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Marek T. Malinowski
TI - Geometrically strictly semistable laws as the limit laws
JO - Discussiones Mathematicae Probability and Statistics
PY - 2007
VL - 27
IS - 1-2
SP - 79
EP - 97
AB - 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.
LA - eng
KW - infinite divisibility; geometric infinite divisibility; geometric semistability; random sums; limit laws; characteristic function
UR - http://eudml.org/doc/277030
ER -

References

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  2. [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. Zbl0579.60016
  3. [3] P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris 1937. Zbl63.0490.04
  4. [4] G.D. Lin, Characterizations of the Laplace and related distributions via geometric compound, Sankhya Ser. A 56 (1994), 1-9. Zbl0806.62010
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  6. [6] M. Maejima, Semistable distributions, in: Lévy Processes, Birkhäuser, Boston 2001, 169-183. 
  7. [7] M. Maejima and G. Samorodnitsky, Certain probabilistic aspects of semistable laws, Ann. Inst. Statist. Math. 51 (1999), 449-462. Zbl0954.60002
  8. [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. Zbl0803.60017
  9. [9] S.T. Rachev and G. Samorodnitsky, Geometric stable distributions in Banach spaces, J. Theoret. Probab. 2 (1994), 351-373. Zbl0804.60014
  10. [10] K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Univ. Press, Cambridge 1999. Zbl0973.60001

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