  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} \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 $S_n$ has
uniformly bounded variance. Here, we give bounds on the moment-generating
functions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have
implications for the inference of evolutionary trees.
