Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays

TL;DR

This paper establishes explicit conditions under which point processes derived from stationary arrays of random variables converge, enhancing understanding of their asymptotic behavior in probabilistic models.

Contribution

It provides new explicit criteria for the convergence of point processes associated with stationary arrays under asymptotic dependence conditions.

Findings

Derived explicit convergence conditions for point processes

Linked convergence to probabilistic behavior of array variables

Applicable to arrays with asymptotic dependence

Abstract

In this article, we consider a stationary array $(X_{j,n})_{1 \leq j \leq n, n \geq 1}$ of random variables with values in $\bR \verb2\2 \{0\}$ (which satisfy some asymptotic dependence conditions), and the corresponding sequence $(N_{n})_{n\geq 1}$ of point processes, where $N_{n}$ has the points $X_{j,n}, 1\leq j \leq n$. Our main result identifies some explicit conditions for the convergence of the sequence $(N_{n})_{n \geq 1}$, in terms of the probabilistic behavior of the variables in the array.