Reversibility is a suucient but not necessary condition for Markov chains for use in Markov chain Monte Carlo simulation. It is necessary to select a Markov chain that has a pre-speciied distribution as its unique stationary distribution. There are many Markov chains that have such property. We give guidelines on how to rank them based on the asymptotic variance of the estimates they produce. T...

(a) If X − Y − Z −W is a Markov chain, then X − Y − Z and Y − Z −W are Markov chains. (b) If X − Y − Z and Y − Z −W are Markov chains, then X − Y − Z −W is a Markov chain. (c) If P (x, y, z, w) = P (x)p(y|x)P (z, w), then X − Y − (Z,W ) is a Markov chain. (d) If X ⊥ Y and Y ⊥ Z, then X ⊥ Z. (e) If the conditional distribution P (x|y, z) is a deterministic function of (x, y), then X − Y − Z is a...

Motivated by an original on-line page-ranking algorithm, starting from an arbitrary Markov chain (Cn) on a discrete state space S, a Markov chain (Cn, Mn) on the product space S, the cat and mouse Markov chain, is constructed. The first coordinate of this Markov chain behaves like the original Markov chain and the second component changes only when both coordinates are equal. The asymptotic pro...

This paper presents a complete framework for the testing procedure based on statistical theory of Markov chains. First, based on this methodology, an analytical evaluation of the Markov representation of time series is undertaken. It is shown that there is a one-to-one correspondence between the AR(1) parameter and the transition probability matrix of the Markov representation of that series. U...

Given a strongly stationary Markov chain and a finite set of stopping rules, we prove the existence of a polynomial algorithm which projects the Markov chain onto a minimal Markov chain without redundant information. Markov complexity is hence defined and tested on some classical problems.

Reversible Markov chains are the basis of many applications. However, computing transition probabilities by a finite sampling of a Markov chain can lead to truncation errors. Even if the original Markov chain is reversible, the approximated Markov chain might be non-reversible and will lose important properties, like the real valued spectrum. In this paper, we show how to find the closest rever...

A lumping of a Markov chain is a coordinate-wise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from t...

We aim at tackling the problem of inadequate specification a Markov manpower model in this paper, by formulating procedure for validating inclusion or non-inclusion some transition parameters model. The mover-stayer principle and its extensions are employed to incorporate hidden classes achieve more homogeneity is compared with without classes, which parsimonious, using Likelihood ratio statist...