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In this paper we provide a complete characterization of the limiting distribution of $T(K_{1, r}, G_n)$, in the regime where $\\mathbb E(T(K_{1, r}, G_n))$ is bounded, for any growing sequence of graphs $G_n$. The asymptotic distribution is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree at most $r$. Conversely, any limiting distribution of $T(K_{1, r}, G_n)$ has a rep"},"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":"1704.04674","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-04-15T18:54:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"73ce1a4f7501636abab341890db2681f40dc610c20ccabd4b4003209066876c1","abstract_canon_sha256":"6fd30d30bbcb03f2cc95cdfa7be4dd102f5814f63de7c3df24d2b1069f2055ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:13.377844Z","signature_b64":"8RnlQ4j5Oh8wISA/aIWSwyRNuvBjVPycqP7QVLJeM+vULbIVaPHgeugwxrY0N/5SjCV5UKzWL8Dyhmcun53eBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a6a291f693b6f68c879cdcea896f1b22eb2d292f224722d46fcf5324a7684bf","last_reissued_at":"2026-05-18T00:29:13.377199Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:13.377199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Limit Theorems for Monochromatic Stars","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Bhaswar B. Bhattacharya, Sumit Mukherjee","submitted_at":"2017-04-15T18:54:50Z","abstract_excerpt":"Let $T(K_{1, r}, G_n)$ be the number of monochromatic copies of the $r$-star $K_{1, r}$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we provide a complete characterization of the limiting distribution of $T(K_{1, r}, G_n)$, in the regime where $\\mathbb E(T(K_{1, r}, G_n))$ is bounded, for any growing sequence of graphs $G_n$. The asymptotic distribution is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree at most $r$. 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