{"paper":{"title":"A Compound Poisson Convergence Theorem for Sums of $m$-Dependent Variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"P. Vellaisamy, V. Cekanavicius","submitted_at":"2014-02-02T12:24:46Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0183","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}