{"paper":{"title":"Surprise probabilities in Markov chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alex Zhai, James Norris, Yuval Peres","submitted_at":"2014-08-04T21:39:01Z","abstract_excerpt":"In a Markov chain started at a state $x$, the hitting time $\\tau(y)$ is the first time that the chain reaches another state $y$. We study the probability $\\mathbf{P}_x(\\tau(y) = t)$ that the first visit to $y$ occurs precisely at a given time $t$. Informally speaking, the event that a new state is visited at a large time $t$ may be considered a \"surprise\". We prove the following three bounds:\n  1) In any Markov chain with $n$ states, $\\mathbf{P}_x(\\tau(y) = t) \\le \\frac{n}{t}$.\n  2) In a reversible chain with $n$ states, $\\mathbf{P}_x(\\tau(y) = t) \\le \\frac{\\sqrt{2n}}{t}$ for $t \\ge 4n + 4$.\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0822","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"}