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We prove the Simons–Johnson theorem for sums \(S_n\) of \(m\) -dependent random variables with exponential weights and limiting compound Poisson distribution \(\mathrm {CP}(s,\lambda )\) . More precisely, we give sufficient conditions for \(\sum _{k=0}^\infty {\mathrm e}^{hk}\vert P(S_n=k)-\mathrm {CP}(s,\lambda )\{k\}\vert \rightarrow 0\) and provide an estimate on the rate of convergence. It is shown that the Simons–Johnson theorem holds for the weighted Wasserstein norm as well. The results are then illustrated for \(N(n;k_1,k_2)\) and \(k\) -runs statistics. Keywords Poisson distribution Compound Poisson distribution M-dependent variables Wasserstein norm Rate of convergence
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