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It turns out that this result can be extended to sums of the form $S_n=\\sum_{i=1}^n\\xi_{q_1(i)}... \\xi_{q_\\ell(i)}$ where now $p_n\\equiv(\\la/n)^{1/\\ell}$ and $1\\leq q_1(i) <... <q_\\ell(i)$ are integer valued increasing functions. We obtain also Poissonian limit for numbers of arrivals to small sets of $\\ell$-tuples $X_{q_1(i)},...,X_{q_\\el"},"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":"1110.2155","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-10T19:35:51Z","cross_cats_sorted":[],"title_canon_sha256":"40415f59014776ac7b9ec5f0b84862fb6b5071bdcb61e9717d6382c0d9f895ce","abstract_canon_sha256":"79bb52885b33b4c4f6b4fb6f1575dce4a56d0b72c607dc63c24f933e3d030ebb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:18.836443Z","signature_b64":"cZOEX+RjfCJnBkoR7hFeuQaAGk0wE51Af+MlUyLJsD/+sF4AwK5tWQrmaZGMnOWjLvGUdcUzFLvUBe057zS3AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a3afb402bd777d50e445e156011a3bd81c3def08a03cac9a9ba1d5d40f3ced0","last_reissued_at":"2026-05-18T04:11:18.835685Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:18.835685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonconventional Poisson Limit Theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yuri Kifer","submitted_at":"2011-10-10T19:35:51Z","abstract_excerpt":"The classical Poisson theorem says that if $\\xi_1,\\xi_2,...$ are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability $p_n\\equiv \\la/n$ then the sum $S_n=\\sum_{i=1}^n\\xi_i$ is asymptotically in $n$ Poisson distributed with the parameter $\\la$. It turns out that this result can be extended to sums of the form $S_n=\\sum_{i=1}^n\\xi_{q_1(i)}... \\xi_{q_\\ell(i)}$ where now $p_n\\equiv(\\la/n)^{1/\\ell}$ and $1\\leq q_1(i) <... <q_\\ell(i)$ are integer valued increasing functions. 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