Convergence of Point Processes with Weakly Dependent Points |
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Authors: | Raluca M Balan Sana Louhichi |
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Affiliation: | 1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada 2. Laboratoire de Probabilités, Statistique et Modélisation, Université de Paris-Sud, Bat. 425, 91405, Orsay Cedex, France
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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}}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
Nn=?j=1ndXj,nN_{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. |
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Keywords: | |
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