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Register a new userQuantitative bounds on convergence of time-inhomogeneous Markov chains
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Convergence rates of Markov chains have been widely studied in recent years. In particular, quantitative bounds on convergence rates have been studied in various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994) 981-1101], Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566], Roberts and Tweedie [Stochastic Process. Appl. 80 (1999) 211-229], Jones and Hobert [Statist. Sci. 16 (2001) 312-334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In this paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566] that concerns quantitative convergence rates for time-homogeneous Markov chains. Our extension allows us to consider f-total variation distance (instead of total variation) and time-inhomogeneous Markov chains. We apply our results to simulated annealing.
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This work contributes to the theory of Wiener-Hopf type factorization for finite Markov chains. This theory originated in the seminal paper Barlow et al. (1980), which treated the case of finite time-homogeneous Markov chains. Since then, several works extended the results of Barlow et al. (1980) in many directions. However, all these extensions were dealing with time-homogeneous Markov case. The first work dealing with the time-inhomogeneous situation was Bielecki et al. (2018), where Wiener-Hopf type factorization for time-inhomogeneous finite Markov chain with piecewise constant generator matrix function was derived. In the present paper we go further: we derive and study Wiener-Hopf type factorization for time-inhomogeneous finite Markov chain with the generator matrix function being a fairly general matrix valued function of time.
Continuous-time Markov chains are mathematical models that are used to describe the state-evolution of dynamical systems under stochastic uncertainty, and have found widespread applications in various fields. In order to make these models computationally tractable, they rely on a number of assumptions that may not be realistic for the domain of application; in particular, the ability to provide exact numerical parameter assessments, and the applicability of time-homogeneity and the eponymous Markov property. In this work, we extend these models to imprecise continuous-time Markov chains (ICTMCs), which are a robust generalisation that relaxes these assumptions while remaining computationally tractable. More technically, an ICTMC is a set of precise continuous-time finite-state stochastic processes, and rather than computing expected values of functions, we seek to compute lower expectations, which are tight lower bounds on the expectations that correspond to such a set of precise models. Note that, in contrast to e.g. Bayesian methods, all the elements of such a set are treated on equal grounds; we do not consider a distribution over this set. The first part of this paper develops a formalism for describing continuous-time finite-state stochastic processes that does not require the aforementioned simplifying assumptions. Next, this formalism is used to characterise ICTMCs and to investigate their properties. The concept of lower expectation is then given an alternative operator-theoretic characterisation, by means of a lower transition operator, and the properties of this operator are investigated as well. Finally, we use this lower transition operator to derive tractable algorithms (with polynomial runtime complexity w.r.t. the maximum numerical error) for computing the lower expectation of functions that depend on the state at any finite number of time points.
Let $X_1,X_2,ldots$ and $Y_1,Y_2,ldots$ be two random sequences so that every random variable takes values in a finite set $mathbb{A}$. We consider a global similarity score $L_n:=L(X_1,ldots,X_n;Y_1,ldots,Y_n)$ that measures the homology (relatedness) of words $(X_1,ldots,X_n)$ and $(Y_1,ldots,Y_n)$. A typical example of such score is the length of the longest common subsequence. We study the order of central absolute moment $E|L_n-EL_n|^r$ in the case where two-dimensional process $(X_1,Y_1),(X_2,Y_2),ldots$ is a Markov chain on $mathbb{A}times mathbb{A}$. This is a very general model involving independent Markov chains, hidden Markov models, Markov switching models and many more. Our main result establishes a general condition that guarantees that $E|L_n-EL_n|^rasymp n^{rover 2}$. We also perform simulations indicating the validity of the condition.
This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices $N_0^d$, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular in biology. We characterize the structure of the state space of general CTMCs on $N_0^d$ in terms of the set of jump vectors and their corresponding transition rate functions. For CTMCs on $N_0$ with polynomial transition rate functions, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, certain absorption, positive/null recurrence, implosivity, and existence and non-existence of moments of hitting times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained, and the few gap cases are well-illustrated by examples. Subtle differences in conditions for different dynamical properties are revealed in terms of examples. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes which are not birth-death processes.
We study certain properties of the function space of autocorrelation functions of Unit Continuous Time Markov Chains (CTMCs). It is shown that under particular conditions, the $L^p$ norm of the autocorrelation function of arbitrary finite state space CTMCs is infinite. Several interesting inferences are made for point processes associated with CTMCs/ Discrete Time Markov Chains (DTMCs).
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