{"paper":{"title":"Rate of convergence of the mean for sub-additive ergodic sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Antonio Auffinger, Jack Hanson, Michael Damron","submitted_at":"2014-08-19T19:55:59Z","abstract_excerpt":"For sub-additive ergodic processes $\\{X_{m,n}\\}$ with weak dependence, we analyze the rate of convergence of $\\mathbb{E}X_{0,n}/n$ to its limit $g$. We define an exponent $\\gamma$ given roughly by $\\mathbb{E}X_{0,n} \\sim ng + n^\\gamma$, and, assuming existence of a fluctuation exponent $\\chi$ that gives $\\mathrm{Var}~X_{0,n} \\sim n^{2\\chi}$, we provide a lower bound for $\\gamma$ of the form $\\gamma \\geq \\chi$. The main requirement is that $\\chi \\neq 1/2$. In the case $\\chi=1/2$ and under the assumption $\\mathrm{Var}~X_{0,n} = O(n/(\\log n)^\\beta)$ for some $\\beta>0$, we prove $\\gamma \\geq \\chi "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4444","kind":"arxiv","version":2},"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"}