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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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