  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.
