Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators |
| 2. | Creator | Author's name, affiliation, country | Yuval Peres; Microsoft |
| 2. | Creator | Author's name, affiliation, country | Sebastien Roch; UCLA |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | Markov chains on trees, reconstruction problem, Kesten-Stigum bound, phylogenetic reconstruction |
| 3. | Subject | Subject classification | 60J80, 92D15 |
| 4. | Description | 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. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | NSF |
| 7. | Date | (YYYY-MM-DD) | 2011-05-19 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ecp.ejpecp.org/article/view/1630 |
| 10. | Identifier | Digital Object Identifier | 10.1214/ECP.v16-1630 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Communications in Probability; Vol 16 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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