{"paper":{"title":"Convergence of a Second Order Markov Chain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.NA","authors_text":"Liqun Qi, Shenglong Hu","submitted_at":"2013-07-26T04:29:52Z","abstract_excerpt":"In this paper, we consider convergence properties of a second order Markov chain. Similar to a column stochastic matrix is associated to a Markov chain, a so called {\\em transition probability tensor} $P$ of order 3 and dimension $n$ is associated to a second order Markov chain with $n$ states. For this $P$, define $F_P$ as $F_P(x):=Px^{2}$ on the $n-1$ dimensional standard simplex $\\Delta_n$. If 1 is not an eigenvalue of $\\nabla F_P$ on $\\Delta_n$ and $P$ is irreducible, then there exists a unique fixed point of $F_P$ on $\\Delta_n$. In particular, if every entry of $P$ is greater than $\\frac{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6919","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"}