Persi Diaconis, Jimmy He, I. Martin Isaacs
Squaring and adding $\pm 1$ mod p generates a curiously intractable random walk. A similar process over the finite field $\mathbf{F}_q$ (with $q=2^d$) leads to novel connections between elementary Galois theory and probability.
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A Markov Chain Arising from the Hopf Square Map on a Non-cocommutative Quantum Group
本文在非交换非余交换量子群上利用Hopf平方映射构造马尔可夫链，扩展了Diaconis等人的工作，并用鞅理论证明了相变及期望增长率界。
This article presents a new proof of the rate of convergence to the normal distribution of sums of independent, identically distributed random variables in chi-square distance, which was also recently studied in \cite{BobkovRenyi}. Our method consists of taking advantage of the underlying time non-homogeneous Markovian structure and studying the spectral properties of the non-reversible transition operator, which allows to find the optimal rate in the convergence above under matching moments assumptions. Our main assumption is that the random variables involved in the sum are independent and h…
Least squares estimation of the transition density in bifurcating Markov models
本文提出一种估计分叉马尔可夫模型转移密度的最小二乘方法，避免了核估计的商误差，并发展指数不等式证明其收敛速度依赖于遍历率，最后以数值研究验证理论结果。
Logarithmic Sobolev inequalities are a fundamental class of inequalities that play an important role in information theory. They play a key role in establishing concentration inequalities and in obtaining quantitative estimates on the convergence to equilibrium of Markov processes. More recently, deep links have been established between logarithmic Sobolev inequalities and strong data processing inequalities. In this paper we study logarithmic Sobolev inequalities from a computational point of view. We describe a hierarchy of semidefinite programming relaxations which give certified lower boun…
Chaotic and Predictable Representations for Markov Additive Processes with Levy Modulator
本文研究了具有莱维调制器的马尔可夫加性过程的混沌和可预测表示，通过正交化Teugels鞅来推导平方可积随机变量和鞅的表示方法。
In this paper we study the asymptotic behavior of linear processes having as innovations mean zero, square integrable functions of stationary reversible Markov chains. In doing so we shall preserve the generality of coefficients assuming only that they are square summable. In this way we include in our study the long range dependence case. The only assumption imposed on the innovations is the absolute summability of their covariances. Besides the central limit theorem we also study the convergence to fractional Brownian motion. The proofs are based on general results for linear processes with …
The Laplace transform of partial sums of the square of a non-centered Gauss-Markov process, conditioning on its starting point, is explicitly computed. The parameters of multiplicative ergodicity are deduced.
Operads and the Markov Property on the square lattice
本文利用算子代数语言为任意维度的格点马尔可夫过程提供了完整的代数描述，提出了边界条件的新参数化方法，并利用Kolmogorov扩张构造了平移不变的无限体积Gibbs测度。
This paper concerns error bounds for recursive equations subject to Markovian disturbances. Motivating examples abound within the fields of Markov chain Monte Carlo (MCMC) and Reinforcement Learning (RL), and many of these algorithms can be interpreted as special cases of stochastic approximation (SA). It is argued that it is not possible in general to obtain a Hoeffding bound on the error sequence, even when the underlying Markov chain is reversible and geometrically ergodic, such as the M/M/1 queue. This is motivation for the focus on mean square error bounds for parameter estimates. It is s…
For a relatively large class of well-behaved absorbing (or killed) finite Markov chains, we give detailed quantitative estimates regarding the behavior of the chain before it is absorbed (or killed). Typical examples are random walks on box-like finite subsets of the square lattice $\mathbb Z^d$ absorbed (or killed) at the boundary. The analysis is based on Poincaré, Nash, and Harnack inequalities, moderate growth, and on the notions of John and inner-uniform domains.
We show that for a general Markov generator the associated square-field (or carré du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the square-field operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operator-theoretic normality condition, th…
We propose some backward-forward martingale decompositions for functions of reversible Markov chains. These decompositions are used to prove the functional CLT for reversible Markov chains with asymptotically linear variance of partial sums. We also provide a proof of the equivalence between asymptotic linearity of the variance and convergence of the integral of $1/(1-t)$ with respect to the associated spectral measure $ρ$. We also study the asymptotic behavior of linear processes having as innovations mean zero square integrable functions of stationary reversible Markov chains. We apply this …
We show how to map the states of an ergodic Markov chain to Euclidean space so that the squared distance between states is the expected commuting time. We find a minimax characterization of commuting times, and from this we get monotonicity of commuting times with respect to equilibrium transition rates. All of these results are familiar in the case of time-reversible chains, where techniques of classical electrical theory apply. In presenting these results, we take the opportunity to develop Markov chain theory in a `conformally correct' way
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 uni…
For an indecomposable $3\times 3$ stochastic matrix (i.e., 1-step transition probability matrix) with coinciding negative eigenvalues, a new necessary and sufficient condition of the imbedding problem for time homogeneous Markov chains is shown by means of an alternate parameterization of the transition rate matrix (i.e., intensity matrix, infinitesimal generator), which avoids calculating matrix logarithm or matrix square root. In addition, an implicit description of the imbedding problem for the $3\times 3$ stochastic matrix in Johansen [J. Lond. Math. Soc., 8, 345-351. (1974)] is pointed ou…
In this paper we study the central limit theorem for additive functionals of stationary Markov chains with general state space by using a new idea involving conditioning with respect to both the past and future of the chain. Practically, we show that any additive functionals of a stationary and totally ergodic Markov chain with the variance of S_n divided by n uniformly bounded, satisfies a square root central limit theorem with a random centering. We do not assume that the Markov chain is irreducible and aperiodic. However, the random centering is not needed if the Markov chain satisfies stro…
We construct strong stationary dual chains for Ising model on a circle, non-symmetric random walk on square lattice and a random walk on hypercube. The strong stationary dual chains are all sharp and have the same state space as original chains. We use Möbius monotonicity of these chains with respect to natural orderings of the corresponding state spaces. This method provides an easy way to find eigenvalues in the Ising model and for a random walk on hypercube. Keywords: Markov chains; stochastic monotonicity; eigenvalues; Möbius monotonicity; strong stationary duality; strong stationary times…
Markov Chain CLTs: Resolving Open Problems
本文解决了Häggström和Rosenthal提出的七个开放问题，证明了可逆链方差极限条件、非紧致性及Roberts条件排除CLT，并建立傅里叶系数与自协方差的一般原理。
It is shown that transient graphs for the simple random walk do not admit a nearest neighbor transient Markov chain (not necessarily a reversible one), that crosses all edges with positive probability, while there is such chain for the square grid $\mathbb{Z}^2$. In particular, the $d$-dimensional grid $\mathbb{Z}^d$ admits such a Markov chain only when $d=2$. For $d=2$ we present a relevant example due to Gady Kozma, while the general statement for transient graphs is obtained by proving that for every transient irreducible Markov chain on a countable state space, which admits a stationary me…
Markov Chain Convergence in Kernel-Induced Geometry
该研究在核诱导几何的收缩假设与谱间隙假设下，建立了马尔可夫链经验占据测度收敛的非渐近高概率界，并将其应用于动力系统一步预测，推导了非线性最小二乘估计器的快速学习率。
Exact random centering and hybrid fluctuation limits for interacting reinforced processes under critical Markov switching
针对临界马尔可夫切换下交互强化过程，提出精确随机中心化与坐标分解，证明混合PDMP-扩散极限并刻画平稳性质，恢复交互信息。
It is often possible to speed up the mixing of a Markov chain $\{ X_{t} \}_{t \in \mathbb{N}}$ on a state space $Ω$ by \textit{lifting}, that is, running a more efficient Markov chain $\{ \hat{X}_{t} \}_{t \in \mathbb{N}}$ on a larger state space $\hatΩ \supset Ω$ that projects to $\{ X_{t} \}_{t \in \mathbb{N}}$ in a certain sense. In [CLP99], Chen, Lov{á}sz and Pak prove that for Markov chains on finite state spaces, the mixing time of any lift of a Markov chain is at least the square root of the mixing time of the original chain, up to a factor that depends on the stationary measure. Unfort…
We investigate the mixing rate of a Markov chain where a combination of long distance edges and non-reversibility is introduced: as a first step, we focus here on the following graphs: starting from the cycle graph, we select random nodes and add all edges connecting them. We prove a square factor improvement of the mixing rate compared to the reversible version of the Markov chain.
Nonlinear Markov Chains (nMC) are regarded as the original (linear) Markov Chains with nonlinear small perturbations. It fits real-world data better, but its associated properties are difficult to describe. A new approach is proposed to analyze the ergodicity and even estimate the convergence bounds of nMC, which is more precise than existing results. In the new method, Coupling Markov about homogeneous Markov chains is applied to reconstitute the relationship between distribution at any times and the limiting distribution. The convergence bounds can be provided by the transition probability m…
Perturbation Analysis of Markov Chain Monte Carlo for Graphical Models
本文研究了图模型中马尔可夫链蒙特卡洛的扰动分析，证明对于图模型相关的目标分布，允许比一般情况大至收敛速率平方根的扰动，且近似MCMC算法中更大的扰动仍能保证较小误差。
Quantum vs classical Markov chains; Exactly solvable examples
本文提出图上可逆马尔可夫链的无硬币量子化方法，通过相似变换构造量子哈密顿量，并基于Askey方案中的正交多项式给出二十多个精确可解例子，以展示量子与经典马尔可夫链的对比。
Operadic structure of boundary conditions for two-dimensional Markov Gaussian random fields on the lattice
本文在二维方格点 Gaussian Markov 场中严格构造并解释 operad 乘积、特征元素及折叠等边界运算，并将其联系到转移矩阵、Fourier 变换与 Schur 补。
Contingency tables are useful objects in statistics for representing 2-way data. With fixed row and column sums, and a total of $n$ entries, contingency tables correspond to parabolic double cosets of $S_n$. The uniform distribution on $S_n$ induces the Fisher-Yates distribution, classical for its use in the chi-squared test for independence. A Markov chain on $S_n$ can then induce a random process on the space of contingency tables through the double cosets correspondence. The random transpositions Markov chain on $S_n$ induces a natural `swap' Markov chain on the space of contingency tables;…
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