Forgetting of the initial distribution for non-ergodic Hidden Markov Chains

序列马尔可夫凝聚过程的遗忘速率

Rates of forgetting for the sequentially Markov coalescent

本文研究了序列马尔可夫凝聚过程在配对设定下的遗忘速率，证明了嵌入跳链的几何平稳性，并展示了连续过程在遗传距离下的平稳距离衰减规律，同时通过新颖的时间变换论证得出了SMC'过程的类似结果。

Stability of the optimal filter in a hidden Markov model with multiplicative noise

We consider a hidden Markov model with multiplicative noise emerging from studies of software reliability. We show the stability of the optimal filter with respect to general initial conditions in the total variation- and $L^p$-norm and deduce explicit rates. Remarkably, stability turns out to be independent of the ergodic behavior of the signal.

Exponential forgetting of smoothing distributions for pairwise Markov models

We consider a bivariate Markov chain $Z=\{Z_k\}_{k \geq 1}=\{(X_k,Y_k)\}_{k \geq 1}$ taking values on product space ${\cal Z}={\cal X} \times{ \cal Y}$, where ${\cal X}$ is possibly uncountable space and ${\cal Y}=\{1,\ldots, |{\cal Y}|\}$ is a finite state-space. The purpose of the paper is to find sufficient conditions that guarantee the exponential convergence of smoothing, filtering and predictive probabilities: $$\sup_{n\geq t}\|P(Y_{t:\infty}\in \cdot|X_{l:n})-P(Y_{t:\infty}\in \cdot|X_{s:n}) \|_{\rm TV} \leq K_s α^{t}, \quad \mbox{a.s.}$$ Here $t\geq s\geq l\geq 1$, $K_s$ is $σ(X_{s:\in…

循环神经网络在ergodic数据序列上的核极限

Kernel Limit for a Class of Recurrent Neural Networks Trained on Ergodic Data Sequences

该研究发展了RNN在ergodic数据序列上的核极限理论，通过固定点分析和微分方程处理，解决了RNN在大规模数据和隐藏单元下的收敛问题。

以二叉树为索引的隐马尔可夫模型最大似然估计量的渐近性质

Asymptotic properties of the maximum likelihood estimator for Hidden Markov Models indexed by binary trees

本文研究以二叉树为索引、隐状态为一般度量空间的隐马尔可夫模型，在平稳与非平稳情形下证明最大似然估计量的强相合性与渐近正态性，并借助树索引马尔可夫链的遍历定理完成证明。

Maximum Likelihood Estimator for Hidden Markov Models in continuous time

The paper studies large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the parameter of a continuous time Markov chain, observed in white noise. Using the method of weak convergence of likelihoods due to I.Ibragimov and R.Khasminskii, consistency, asymptotic normality and convergence of moments are established for MLE under certain strong ergodicity conditions of the chain.

Uniform Chernoff and Dvoretzky-Kiefer-Wolfowitz-type inequalities for Markov chains and related processes

We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of non-contracting chains. In particular, geometric ergodicity provides a simple and versatile framework. This leads to a short, elementary proof of a general concentration inequality for Markov and hidden Markov chains (HMM), which supercedes some of the known results and easily extends to other processes such as Markov trees. As applications, we give a Dvoretzky-Kiefer-Wolfowitz-type inequality and a uniform Chernoff bound. All of our bounds are dimension-free and hold for cou…

The stability of conditional Markov processes and Markov chains in random environments

We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of $σ$-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365--393]. …

隐马尔可夫模型中滤波稳定性的对偶理论

Duality Theory for Filter Stability in Hidden Markov Models

本文针对有限状态离散时间隐马尔可夫模型，建立以倒向随机差分方程为对偶控制系统的对偶理论，推导不变性原理并将其应用于滤波稳定性与收敛速率分析。

滤波器稳定性的方差衰减性质

Variance Decay Property for Filter Stability

本文研究白噪声观测下HMM的非线性滤波器稳定性问题，提出方差衰减性质并引入新的庞加莱不等式概念，基于最小方差对偶性将问题转化为BSDE的随机最优控制问题进行证明。

用于滤波稳定性分析的后向映射

Backward Map for Filter Stability Analysis

本文提出后向映射方法分析非线性随机滤波稳定性，并通过为白噪声观测 HMM 引入庞加莱不等式，证明庞加莱常数为正当且仅当 HMM 可检测。