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Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the infinite $b$-arytree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b \geq 2$,$k \geq 2$ and $[k] = \{1,...,k\}$. We denote by $L_n$ the level-$n$ verticesof $T$. 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 wasintroduced by Mossel and Peres (2003) in the context of the "recontructionproblem" on trees: \begin{equation*} S_n = (b\lambda)^{-n} \sum_{x\in L_n}\sigma_x. \end{equation*} As noted by Mossel and Peres, when $b\lambda^2 > 1$(the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$ hasuniformly bounded variance. Here, we give bounds on the moment-generatingfunctions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results haveimplications for the inference of evolutionary trees.

Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators