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It's known that $\\{X_n\\}$ is positive recurrent when $\\mu<1$; is null recurrent when $\\mu=1$; and is transient when $\\mu>1$. In this paper, we shall discuss the first returning speed and the last exit speed more precisely by means of $\\{a_n\\}$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1101.1161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-06T08:19:55Z","cross_cats_sorted":[],"title_canon_sha256":"bfaa97564c833e654a39fa19a40596e75adddbd7c498bfc7e4dff8a522007762","abstract_canon_sha256":"22e7015cc4ddd180bb6a80388d6377d7722509b68e53786860ddedc14704a54d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:00.419080Z","signature_b64":"MKTVodb54JUyqBvhbXwBU1E7i+eogrdd4zOaays5YhQZAoPdyw+XdB3K1V9arzKjz+UHa8EEcMdlB1mWCNRODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d870b2c3792b29d7937a86595b63a4261015bd6d378c9d329d91ab6005b403e2","last_reissued_at":"2026-05-18T04:32:00.418623Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:00.418623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The first returning speed and the last exit speed of a type of Markov chain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Huizeng Zhang, Lei Wang, Minzhi Zhao","submitted_at":"2011-01-06T08:19:55Z","abstract_excerpt":"Let $\\{X_n\\}$ be a Markov chain with transition probability $p_{ij}=a_{j-(i-1)^+},\\forall i,j\\ge 0$, where $a_j=0$ provided $j<0$, $a_0>0$, $a_0+a_1<1$ and $\\sum_{n=0}^\\infty a_n=1$. Let $\\mu=\\sum_{n=1}^\\infty na_n$. It's known that $\\{X_n\\}$ is positive recurrent when $\\mu<1$; is null recurrent when $\\mu=1$; and is transient when $\\mu>1$. 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