{"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. We obtain also Poissonian limit for numbers of arrivals to small sets of $\\ell$-tuples $X_{q_1(i)},...,X_{q_\\el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2155","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"}