Convergence of Point Processes with Weakly Dependent Points

Raluca M. Balan () and Sana Louhichi ()
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Raluca M. Balan: University of Ottawa
Sana Louhichi: Université de Paris-Sud

Journal of Theoretical Probability, 2009, vol. 22, issue 4, 955-982

Abstract: 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.

Keywords: Weak limit theorem; Point process; Infinitely divisible law; Strong mixing; Association; 60F05; 60E07 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0176-4

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