{"paper":{"title":"Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.ST","q-bio.PE","stat.TH"],"primary_cat":"math.PR","authors_text":"Sebastien Roch, Yuval Peres","submitted_at":"2009-08-14T13:05:32Z","abstract_excerpt":"Consider a Markov chain $(\\xi_v)_{v \\in V} \\in [k]^V$ on the infinite $b$-ary tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b \\geq 2$, $k \\geq 2$ and $[k] = \\{1,...,k\\}$. We denote by $L_n$ the level-$n$ vertices of $T$. Assume $M$ has a real second-largest (in absolute value) eigenvalue $\\lambda$ with corresponding real eigenvector $\\nu \\neq 0$. Letting $\\sigma_v = \\nu_{\\xi_v}$, we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the \"recontruction problem\" on trees: \\begin{equation*} S_n = (b\\lambda)^{-n} \\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.2056","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"}