Covariance and relaxation time in finite Markov chains (Q1271257)

Summary: The relaxation time \(T_{\text{REL}}\) of a finite ergodic Markov chain in continuous time, i.e., the time to reach ergodicity from some initial state distribution, is loosely given in the literature in terms of the eigenvalues \(\lambda_j\) of the infinitesimal generator \({\mathbf Q}\). One uses \(T_{\text{REL}}= \theta^{-1}\) where \(\theta= \min_{\lambda_j\neq 0} \{\text{Re }\lambda_j[- {\mathbf Q}]\}\). This paper establishes for the relaxation time \(\theta^{-1}\) the theoretical solidity of the time reversible case. It does so by examining the structure of the quadratic distance \(d(t)\) to ergodicity. It is shown that, for any function \(f(j)\) defined for states \(j\), the correlation function \(\rho_f(\tau)\) has the bound \(|\rho_f(\tau)|\leq \exp[-\theta|\tau|]\) and that this inequality is tight. The argument is almost entirely in the real domain.