Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators

Consider a Markov chain (ξv)v∈V ∈ [k] on the infinite b-ary tree T = (V,E) with irreducible edge transition matrix M , where b ≥ 2, k ≥ 2 and [k] = {1, . . . , k}. We denote by Ln the level-n vertices of T . Assume M has a real second-largest (in absolute value) eigenvalueλ with corresponding real eigenvector ν 6= 0. Letting σv = νξv , we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the “recontruction problem” on trees: