Taiwanese Journal of Mathematics

STABLE POISSON CONVERGENCE FOR INTEGER-VALUED RANDOM VARIABLES

Tsung-Lin Cheng and Shun-Yi Yang

Full-text: Open access

Abstract

In this paper, we obtain some stable Poisson Convergence Theorems for arrays of integer-valued dependent random variables. We prove that the limiting distribution is a mixture of Poisson distribution when the conditional second moments on a given $\sigma$-algebra of the sequence converge to some positive random variable. Moreover, we apply the main results to the indicator functions of rowise interchangeable events and obtain some interesting stable Poisson convergence theorems.

Article information

Source
Taiwanese J. Math., Volume 17, Number 6 (2013), 1869-1885.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706275

Digital Object Identifier
doi:10.11650/tjm.17.2013.1751

Mathematical Reviews number (MathSciNet)
MR3141864

Zentralblatt MATH identifier
1291.60044

Subjects
Primary: 60F05: Central limit and other weak theorems

Keywords
stable Poisson Convergence

Citation

Cheng, Tsung-Lin; Yang, Shun-Yi. STABLE POISSON CONVERGENCE FOR INTEGER-VALUED RANDOM VARIABLES. Taiwanese J. Math. 17 (2013), no. 6, 1869--1885. doi:10.11650/tjm.17.2013.1751. https://projecteuclid.org/euclid.twjm/1499706275


Export citation

References

  • R. J. Adler and D. J. Scott, Martingale central limit theorems without negligibility conditions, Corrigendum. Bull. Austral. Math. Soc., 18 (1978), 311-319.
  • D. J. Aldous and G. K. Eagleson, On mixing and stability of limit theorems, Ann. Probab., 6 (1978), 325-331.
  • D. J. Aldous, Exchangeability and Related Topics, Lecture Notes in Mathematics, Springer-Verlag, 1983.
  • A. D. Barbour and L. Holst, Some applications of the Stein-Chen method for proving Poisson convergence, Adv. Appl. Prob., 21 (1989), 74-90.
  • S. Bernstein, Sur l'extension du théoréme limite du calcul des probabilitiés aux sommes de quantités dépendantes, Math. Ann., 85 (1927), 1-59.
  • M. Be\ptmr ška, A. Klopotowski and L. Slomi\ptmr šski, Limit Theorems for Random Sums of Dependent d-Dimensional Random Vectors, Z. Wahrscheinlichkeitsth, 61 (1982), 43-57.
  • P. Billingsley, The Lindeberg-Lévy theorem for martingales, Proc. Amer. Math. Soc., 12 (1961b), 788-792.
  • B. Brown, A martingale approach to the Poisson convergence of simple point processes, Ann. Prob., 6 (1978), 615-628.
  • B. Brown and G. K. Eagleson, Martingale convergence to infinitely divisible laws with finite variance, Trans. Amer. Math. Soc., 162 (1971), 449-453.
  • B. Brown, Martingale central limit theorems, Ann. Math. Statist., 42 (1971), 59-66.
  • L. H. Y. Chen, On the convergence of Poisson binomial to Poisson distributions, Ann. Prob., 2 (1974), 178-180.
  • L. H. Y. Chen, Poisson approximation for dependent trials, Ann. Prob., 3 (1975), 534-545.
  • T. L. Cheng and Y. S. Chow, On stable convergence in the central limit theorem, Statist. Probab. Lett., 57 (2002), 307-313.
  • Y. S. Chow and H. Teicher, Probability Theory, 3rd Edition, Springer, Berlin, 1997.
  • J. L. Doob, Stochastic Processes, Wiley, New York, 1953.
  • R. Durrent, Probability: Theory and Examples, Wadsworth and Brooks/Cole, Belmont, CA, 1991.
  • A. Dvoretzky, Central limit theorems for dependent random variables and some applications, Abstract. Ann. Math. Statist, 40 (1969), 1871.
  • A. Dvoretzky, The central limit problem for dependent random variables. in: Actes du Congres International deś Mathematiciens, Nice, 565-570, Gauthier-Villars, Paris, 1971.
  • A. Dvoretzky, Asymptotic normality for sums of dependent random variables, Proc. Sixth Berkeley Symp. Math. Statist, Probability, Univ. of California Pess, 1972, pp. 513-535.
  • G. K. Eagleson, Martingale convergence to the Poisson distribution, Casopis Pest. Mat., 101 (1976), 271-277.
  • G. K. Eagleson, A Poisson Limit Theorem for Weakly Exchangeable Events, J. Appl. Prob., 16 (1979), 794-802.
  • G. K. Eagleson, Weak limit theorems for exchangeable random variables, in: Exchangeability in Probability and Statistics, G. Koch and F. Spizzichino, eds., North-Holland, Amsterdam-New York, 1982, pp. 251-268.
  • D. Freedman, The Poisson approximation for dependent events, Ann. Prob., 2 (1974), 256-269.
  • P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.
  • I. S. Helland, Central limit theorems for martingales with discrete or continuous time, Scand. J. Statist, 9 (1982), 79-94.
  • I. A. Ibragimov, A central limit theorem for a class of dependent random variables, Theory Probab. Appl., 8 (1963), 83-89.
  • N. Kaplan, Two applications of a Poisson approximation for dependent events, Ann. Prob., 5 (1977), 787-794.
  • R. G. Laha and V. K. Rohagi, Probability Theory, 1970.
  • P. Lévy, Propriétés asymptotiques des sommes de variables aléatoires enchainées, Bull. Sci. Math., 59(ser.2) (1935a), 84-96, 109-128.
  • P. Lévy, Propriétés asymptotiques des sommes de variables aléatoires independantes ou enchainées, J. Math. Pures Appl., bf 14(ser.9) (1935b), 347-402.
  • P. Lévy, Théorie de l'adition des Variables Aléatoires, Gauthier-Villars, Paris, 1937.
  • R. M. Loynes, The central limit theorem for backwards martingales, Z. Wahrsch. Verw. Gebiete., 13 (1969), 1-8.
  • D. L. McLeish, Dependent central limit theorem and invariance principles, Ann. Probab., 2 (1974), 620-628.
  • A. Rényi, On stable sequences of events, Sankhyā Ser. A, 25 (1963), 293-302.
  • A. Rényi, Foundations of Probability, Holden-Day, San Francisco, 1970.
  • B. Rosén, On the central limit theorem for sums of dependent random variables, Z. Wahrsch. Verw. Gebiete., 7 (1967a), 48-52.
  • Rosén, B. On asymptotic normality of sums of dependent random variables, Z. Wahrsch. Verw. Gebiete, 7 (1967b), 95-102.
  • A. N. Shiryayev, Probability, Springer, Berlin, New York, 1984.
  • A. G. Sholomitskii, On the necessary conditions of normal convergence for martingales, Theory Probab. Appl., 43 (1998), 434-448.
  • A. G. Sholomitskii, On the necessary conditions of poisson convergence for martingales, Theory Probab. Appl., 49 (2005), 735-737.
  • B. W. Silverman and T. C. Brown, Short distances, flat triangles and Poisson limits, J. Appl. Prob., 15 (1978), 815-825.
  • N. C. Weber, A Martingale Approach to Central Limit Theorems for Exchangeable Random Variables, J. Appl. Prob., 17 (1980), 662-673.
  • J. Wesollowski, Poisson Process via Martingale and Related Characteristics, J. Appl. Prob., 36 (1999), 919-926.