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Convergence of Point Processes with Weakly Dependent Points

Author

Listed:
  • Raluca M. Balan

    (University of Ottawa)

  • Sana Louhichi

    (Université de Paris-Sud)

Abstract

For each n≥1, let {X j,n }1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}$ to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S n =∑ j=1 n X j,n to an infinitely divisible random variable whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.

Suggested Citation

  • Raluca M. Balan & Sana Louhichi, 2009. "Convergence of Point Processes with Weakly Dependent Points," Journal of Theoretical Probability, Springer, vol. 22(4), pages 955-982, December.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:4:d:10.1007_s10959-008-0176-4
    DOI: 10.1007/s10959-008-0176-4
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    References listed on IDEAS

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    1. Jakubowski, Adam, 1997. "Minimal conditions in p-stable limit theorems -- II," Stochastic Processes and their Applications, Elsevier, vol. 68(1), pages 1-20, May.
    2. Jakubowski, Adam, 1993. "Minimal conditions in p-stable limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 44(2), pages 291-327, February.
    3. Denker, Manfred & Jakubowski, Adam, 1989. "Stable limit distributions for strongly mixing sequences," Statistics & Probability Letters, Elsevier, vol. 8(5), pages 477-483, October.
    4. Dedecker, Jérôme & Louhichi, Sana, 2005. "Conditional convergence to infinitely divisible distributions with finite variance," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 737-768, May.
    5. Kobus, M., 1995. "Generalized Poisson Distributions as Limits of Sums for Arrays of Dependent Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 199-244, February.
    6. Jakubowski, Adam & Kobus, Maria, 1989. "[alpha]-Stable limit theorems for sums of dependent random vectors," Journal of Multivariate Analysis, Elsevier, vol. 29(2), pages 219-251, May.
    7. Resnick, Sidney & Samorodnitsky, Gennady, 2004. "Point processes associated with stationary stable processes," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 191-209, December.
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