{"paper":{"title":"Persistence of Gaussian processes: non-summable correlations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Amir Dembo, Sumit Mukherjee","submitted_at":"2015-08-26T20:49:11Z","abstract_excerpt":"Suppose the auto-correlations of real-valued, centered Gaussian process $Z(\\cdot)$ are non-negative and decay as $\\rho(|s-t|)$ for some $\\rho(\\cdot)$ regularly varying at infinity of order $-\\alpha \\in [-1,0)$. With $I_\\rho(t)=\\int_0^t \\rho(s)ds$ its primitive, we show that the persistence probabilities decay rate of $ -\\log\\mathbb{P}(\\sup_{t \\in [0,T]}\\{Z(t)\\}<0)$ is precisely of order $(T/I_\\rho(T)) \\log I_\\rho(T)$, thereby closing the gap between the lower and upper bounds of \\cite{NR}, which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06659","kind":"arxiv","version":3},"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"}