Convergence of Point Processes with Weakly Dependent Points

2008/05/27 by Raluca M. Balan, Balan, Raluca, Sana Louhichi +1
Computer Science · Decision Sciences · Mathematics · #60E07 #60F05 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0805.4128

openalex publication_date 2008/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=∑_j=1ⁿδ_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=1ⁿX_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.

Citations

• Association of Random Variables, with Applications

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