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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1402.0183","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2014-02-02T12:24:46Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"540d5d8ec6d906d74a476851536bb7d134bf3c35a1de9619d99875140cf2da5e","abstract_canon_sha256":"e9afe92df01f5d3a6bc0a08ccfe080881e20858dbc9b929ede5762531d4b927d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:24.443792Z","signature_b64":"seQQ5YQHlHr9aeEZma1SATQNHN45Fx4NMih1NaIMUUXdnP4UNUoqrVK96ym57gakEOLJ8560BY2oOx5gQEZPAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a7da195fa4d73d6277d2b32839fd53963a406781d5c065f943aca5f776de02f","last_reissued_at":"2026-05-18T03:00:24.443091Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:24.443091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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