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

Abstract

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