标题： 一个关于$m$-相关变量和的复合泊松收敛定理 显示英文标题

标题： A Compound Poisson Convergence Theorem for Sums of $m$-Dependent Variables

摘要： 我们证明了对于依赖于$m$的随机变量和为$S_n$的 Simons-Johnson 定理，其中带有指数权重且具有极限复合泊松分布$\CP(s,\lambda)$。 更具体地说，我们给出了$\sum_{k=0}^\infty\ee^{hk}\ab{P(S_n=k)-\CP(s,\lambda)\{k\}}\to 0$成立的充分条件，并提供了收敛速度的估计值。 结果表明，Simons-Johnson 定理同样适用于带权 Wasserstein 范数。 然后，结果被用于$N(n;k_1,k_2)$和$k$的运行统计量上。 显示英文摘要

摘要： We prove the Simons-Johnson theorem for the sums $S_n$ of $m$-dependent random variables, with exponential weights and limiting compound Poisson distribution $\CP(s,\lambda)$. More precisely, we give sufficient conditions for $\sum_{k=0}^\infty\ee^{hk}\ab{P(S_n=k)-\CP(s,\lambda)\{k\}}\to 0$ and provide an estimate on the rate of convergence. It is shown that the Simons-Johnson theorem holds for weighted Wasserstein norm as well. %limiting sum of two Poisson variables defined on %different lattices. The results are then illustrated for $N(n;k_1,k_2)$ and $k$-runs statistics.