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 Mathematics , 2003, Abstract: Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single vertex. We prove a large deviation principle in $n$, with explicit rate function, for the empirical subtree measures of multitype Galton-Watson trees conditioned to have exactly $n$ vertices. In the process, we extend the notions of shift-invariance and specific relative entropy--as typically understood for Markov fields on deterministic graphs such as $\mathbb Z^d$--to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with $n$ vertices.
 Journal of Inequalities and Applications , 2009, Abstract: We study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the property of the harmonic mean of random transition probability for a nonhomogeneous Markov chain.
 Journal of Inequalities and Applications , 2009, DOI: 10.1155/2009/503203 Abstract: We study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the property of the harmonic mean of random transition probability for a nonhomogeneous Markov chain.
 Huilin Huang International Journal of Stochastic Analysis , 2013, DOI: 10.1155/2013/961571 Abstract: We prove some limit properties of the harmonic mean of a random transition probability for finite Markov chains indexed by a homogeneous tree in a nonhomogeneous Markovian environment with finite state space. In particular, we extend the method to study the tree-indexed processes in deterministic environments to the case of random enviroments. 1. Introduction A tree is a graph which is connected and doesn't contain any circuits. Given any two vertices , let be the unique path connecting and . Define the graph distance to be the number of edges contained in the path . Let be an infinite tree with root . The set of all vertices with distance from the root is called the th generation of , which is denoted by . We denote by the union of the first generations of . For each vertex , there is a unique path from to and for the number of edges on this path. We denote the first predecessor of by . The degree of a vertex is defined to be the number of neighbors of it. If every vertex of the tree has degree , we say it is Cayley’s tree, which is denoted by . Thus, the root vertex has neighbors in the first generation and every other vertex has neighbors in the next generation. For any two vertices and of tree , write if is on the unique path from the root to . We denote by the farthest vertex from satisfying and . We use the notation and denote by the number of vertices of . In the following, we always let denote the Cayley tree . A tree-indexed Markov chain is the particular case of a Markov random field on a tree. Kemeny et al. [1] and Spitzer [2] introduced two special finite tree-indexed Markov chains with finite transition matrix which is assumed to be positive and reversible to its stationary distribution, and these tree-indexed Markov chains ensure that the cylinder probabilities are independent of the direction we travel along a path. In this paper, we omit such assumption and adopt another version of the definition of tree-indexed Markov chains which is put forward by Benjamini and Peres [3]. Yang and Ye[4] extended it to the case of nonhomogeneous Markov chains indexed by infinite Cayley’s tree and we restate it here as follows. Definition 1 (T-indexed nonhomogeneous Markov chains (see [4])). Let be an infinite Cayley tree, a finite state space, and a stochastic process defined on probability space , which takes values in the finite set . Let be a distribution on and a transition probability matrix on . If, for any vertex , then will be called -valued nonhomogeneous Markov chains indexed by infinite Cayley’s tree with initial distribution (1) and
 Mathematics , 2014, Abstract: We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\widetilde{\mathcal Z}^n = (\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$. Precisely, we show that if $\widetilde{\mathcal R}^n \to \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \to \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$.
 Arnaud Durand Mathematics , 2007, DOI: 10.1007/s00220-008-0504-7 Abstract: We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection.
 Siméon Valère Bitseki Penda Mathematics , 2013, Abstract: We provide deviation inequalities for properly normalized sums of bifurcating Markov chains on Galton-Watson tree. These processes are extension of bifurcating Markov chains (which was introduced by Guyon to detect cellular aging from cell lineage) in case the index set is a binary Galton-Watson process. As application, we derive deviation inequalities for the least-squares estimator of autoregressive parameters of bifurcating autoregressive processes with missing data. These processes allow, in case of cell division, to take into account the cell's death. The results are obtained under an uniform geometric ergodicity assumption of an embedded Markov chain.
 Huilin Huang ISRN Applied Mathematics , 2012, DOI: 10.5402/2012/536530 Abstract:
 Mathematics , 2010, DOI: 10.1007/s00023-011-0107-2 Abstract: In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two now quasi equivalent QMC for the given family of interaction operators $\{K_{}\}$.
 Mathematics , 2010, DOI: 10.1142/S021902571100447X Abstract: In the present paper we study forward Quantum Markov Chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction we investigate QMC associated with XY-model on a Caylay tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions.
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