Convergence of Point Processes with Weakly Dependent Points

TL;DR

This paper establishes conditions under which sequences of weakly dependent stationary random variables lead to the convergence of associated point processes and partial sums to infinitely divisible limits, extending classical results to dependent data.

Contribution

It provides new asymptotic weak dependence criteria for point process convergence and links this to the distributional limits of partial sums, including dependent structures like mixing and stochastic volatility.

Findings

Derived weak dependence conditions for point process convergence.

Proved convergence of partial sums to infinitely divisible distributions.

Applied results to models with known dependence structures.

Abstract

For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ 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 $S_n=\sum_{j=1}^{n}X_{j,n}$ to an infinitely divisible random variable, whose L\'{e}vy 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.