On some limit distributions for geometric random sums

Marek T. Malinowski

Discussiones Mathematicae Probability and Statistics (2008)

  • Volume: 28, Issue: 2, page 247-266
  • ISSN: 1509-9423

Abstract

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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 .

How to cite

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Marek T. Malinowski. "On some limit distributions for geometric random sums." Discussiones Mathematicae Probability and Statistics 28.2 (2008): 247-266. <http://eudml.org/doc/277016>.

@article{MarekT2008,
abstract = {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 $ℝ^d$.},
author = {Marek T. Malinowski},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {random sum; infinite divisibility; semistability; geometric infinite divisibility; geometric stability; geometric semistability; characteristic function; limit distribution; Lévy process},
language = {eng},
number = {2},
pages = {247-266},
title = {On some limit distributions for geometric random sums},
url = {http://eudml.org/doc/277016},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Marek T. Malinowski
TI - On some limit distributions for geometric random sums
JO - Discussiones Mathematicae Probability and Statistics
PY - 2008
VL - 28
IS - 2
SP - 247
EP - 266
AB - 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 $ℝ^d$.
LA - eng
KW - random sum; infinite divisibility; semistability; geometric infinite divisibility; geometric stability; geometric semistability; characteristic function; limit distribution; Lévy process
UR - http://eudml.org/doc/277016
ER -

References

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  1. [1] D. Applebaum, Lévy processes and stochastic calculus, Cambridge Univ. Press, Cambridge 2004. Zbl1073.60002
  2. [2] B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for sums of independent random variables, second ed., Addison-Wesley, Reading, Mass.-London 1968. Zbl0056.36001
  3. [3] V. Kalashnikov, Geometric sums: Bounds for rare events with applications, Kluwer Academic Publishers, Dordrecht 1997. Zbl0881.60043
  4. [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. Zbl0579.60016
  5. [5] T.J. Kozubowski, The inner characterization of geometric stable laws, Statist. Decisions 12 (1994), 307-321. Zbl0923.60020
  6. [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. Zbl0853.60009
  7. [7] T.J. Kozubowski and S.T. Rachev, Univariate geometric stable laws, J. Comput. Anal. Appl. 1 (1999), 177-217. 
  8. [8] G.D. Lin, Characterizations of the Laplace and related distributions via geometric compound, Sankhya Ser. A 56 (1994), 1-9. Zbl0806.62010
  9. [9] E. Lukacs, Characteristic functions, second ed., Griffin, London 1970. 
  10. [10] M.T. Malinowski, Geometrically strictly semistable laws as the limit laws, Discussiones Mathematicae Probability and Statistics 27 (2007), 79-97. 
  11. [11] M. Maejima and G. Samorodnitsky, Certain probabilistic aspects of semistable laws, Ann. Inst. Statist. Math. 51 (1999), 449-462. Zbl0954.60002
  12. [12] D. Mejzler, On a certain class of infinitely divisible distributions, Israel J. Math. 16 (1973), 1-19. Zbl0276.60022
  13. [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. Zbl0803.60017
  14. [14] S.T. Rachev and G. Samorodnitsky, Geometric stable distributions in Banach spaces, J. Theoret. Probab. 2 (1994), 351-373. Zbl0804.60014
  15. [15] G. Samorodnitsky and M.S. Taqqu, Stable non-gaussian random processes: stochastic models with infinite variance, Chapman and Hall, New York-London 1994. Zbl0925.60027
  16. [16] K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Univ. Press, Cambridge 1999. Zbl0973.60001

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