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

We prove the Simons-Johnson theorem for the sums $S_n$ of $m$-dependent random variables, with exponential weights and limiting compound Poisson distribution $\CP(s,λ)$. More precisely, we give sufficient conditions for $\sum_{k=0}^\infty\ee^{hk}\ab{P(S_n=k)-\CP(s,λ)\{k\}}\to 0$ 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.