树上重构：线性估计的指数矩界
Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators
摘要：
考虑具有不可约边转移矩阵$M$的无限$b$叉树$T = (V,E)$上的马尔可夫链$(ξ_v)_{v \in V} \in [k]^V$，其中$b \geq 2$，$k \geq 2$和$[k] = \{1,...,k\}$ .我们用$L_n$表示$T$的$n$级顶点.假设$M$具有实第二大（绝对值）特征值$λ$，其具有对应的实特征向量$ν\neq 0$。让$σ_v = ν_{ξ_v}$，我们考虑下面的根状态估计，这是由Mossel和Peres（2003）在树的“重构问题”的上下文中引入的：\begin{equation*} S_n = (bλ)^{-n} \sum_{x\in L_n} σ_x. \end{equation*}正如Mossel和Peres所指出的，当$bλ^2 > 1$（所谓的Kesten-Stigum重构阶段）时，量$S_n$具有一致有界方差。在这里，我们给出了当$S_n^2$。我们的研究结果的推断进化树的影响。
英文摘要：
Consider a Markov chain $(ξ_v)_{v \in V} \in [k]^V$ 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 $λ$ with corresponding real eigenvector $ν\neq 0$. Letting $σ_v = ν_{ξ_v}$, 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 = (bλ)^{-n} \sum_{x\in L_n} σ_x. \end{equation*} As noted by Mossel and Peres, when $bλ^2 > 1$ (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 $bλ^2 > 1$. Our results have implications for the inference of evolutionary trees.
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