## Electronic Communications in Probability

### Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators

#### Abstract

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 $[k] = \{1,\ldots,k\}$. 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.

#### Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 24, 251-261.

Dates
Accepted: 19 May 2011
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465261980

Digital Object Identifier
doi:10.1214/ECP.v16-1630

Mathematical Reviews number (MathSciNet)
MR2802041

Zentralblatt MATH identifier
1237.60068

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D15: Problems related to evolution

Rights

#### Citation

Peres, Yuval; Roch, Sebastien. Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators. Electron. Commun. Probab. 16 (2011), paper no. 24, 251--261. doi:10.1214/ECP.v16-1630. https://projecteuclid.org/euclid.ecp/1465261980

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