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For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ be a sequence ofstrictly stationary random variables. In this article, we give some asymptoticweak dependence conditions for the convergence in distribution of the pointprocess $N_n=\sum_{j=1}^{n}\delta_{X_{j,n}}$ to an infinitely divisible pointprocess. From the point process convergence, we obtain the convergence indistribution of the partial sum sequence $S_n=\sum_{j=1}^{n}X_{j,n}$ to aninfinitely divisible random variable, whose L\'{e}vy measure is related to thecanonical measure of the limiting point process. As examples, we discuss thecase of triangular arrays which possess known (row-wise) dependence structures,like the strong mixing property, the association, or the dependence structureof a stochastic volatility model.

Convergence of Point Processes with Weakly Dependent Points