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

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