Analysis of non-reversible Markov chains via similarity orbit

Abstract

In this paper, we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as the one of birth-death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth-death one. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and ${\mathrm{L}}^2$-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper, we investigate a particular similarity orbit of reversible Markov kernels, that we call the pure birth orbit, and analyze various possibly non-reversible variants of classical birth-death processes in this orbit.

arXiv comment29 pages. To appear in Combin. Probab. Comput

References in corpus (10)

• Separation cut-offs for birth and death chains
• Non-reversible Metropolis-Hastings
• Exactly Solvable Birth and Death Processes
• Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions
• Intertwining and commutation relations for birth-death processes
• The ODE method for stability of skip-free Markov chains with applications to MCMC
• Partial inner product spaces, metric operators and generalized hermiticity
• Skip-free Markov chains
• Convergence analysis of some multivariate Markov chains using stochastic monotonicity
• Gibbs Sampling, Exponential Families and Orthogonal Polynomials