{"paper":{"title":"Central limit theorems for the shrinking target problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Licheng Zhang, Matthew Nicol, Nicolai Haydn, Sandro Vaienti","submitted_at":"2013-05-26T21:43:52Z","abstract_excerpt":"Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,\\mu)$. The question of whether $T^i x\\in B_i$ infinitely often (i. o.) for $\\mu$ a.e.\\ $x$ is often called the shrinking target problem. In many dynamical settings it has been shown that if $E_n:=\\sum_{i=1}^n \\mu (B_i)$ diverges then there is a quantitative rate of entry and $\\lim_{n\\to \\infty} \\frac{1}{E_n} \\sum_{j=1}^{n} 1_{B_i} (T^i x) \\to 1$ for $\\mu$ a.e. $x\\in X$. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6073","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"}