Hidden Markov processes in the context of symbolic dynamics

符号动力学、伪轨跟踪与某些可数分段线性马尔可夫映射的实数表示

Symbolic dynamics, shadowing and representation of real numbers with some countably piecewise linear Markov maps

受2022年施韦策竞赛问题启发，研究可数分段线性马尔可夫映射，引入ℓ-马尔可夫分割并关联符号动力学与马尔可夫转移，证明竞赛系统的伪轨跟踪性质，研究有理数轨道及最终周期性，提供示例系统。

Symbolic Dynamics and Markov Partitions

The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two…

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.

Decomposition of conditional probability for high-order symbolic Markov chains

The main goal of the paper is to develop an estimate for the conditional probability function of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet. We elaborate a decomposition procedure for the conditional probability function of sequences considered as the high-order Markov chains. We represent the conditional probability function as the sum of multi-linear memory function monomials of different orders (from zero up to the chain order). This allows us to construct artificial sequences by method of successive iterations taking into account at each step …

Symbolic dynamics for nonuniformly hyperbolic systems

This survey describes the recent advances in the construction of Markov partitions for nonuniformly hyperbolic systems. One important feature of this development comes from a finer theory of nonuniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes

In this note, we show that small complex perturbations of positive matrices are contractions, with respect to a complex version of the Hilbert metric, on the standard complex simplex. We show that this metric can be used to obtain estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.