Eigentime identity for asymmetric finite Markov chains
An eigentime identity relates certain hitting/return time quantities associated to a Markov chain with the eigenvalues of its transition probability matrix. Possibly the simplest identity of this type is the following. Consider a reversible continuous-time Markov chain on \(E=\{1,\dots,N\}\) with the jump rate matrix \(Q=(q_{ij})\) which is assumed to be irreducible and conservative. Let \(\pi=(\pi_i)\) be the vector of the (unique) invariant probability distribution. Reversibility means that \(\pi_i q_{ij}=\pi_j q_{ji}\). That is, the operator \(Q\) in \(L^2(\pi)\) is self-adjoint. Let \(\lambda_0=0,\lambda_1,\dots,\lambda_N\) be the eigenvalues of \((-Q)\) which are real and nonnegative. Let \(\tau_j\) be the hitting time of \(j\), i.e., \(\tau_j=\inf\{t\geq0: X_t=j\}\). Consider the average hitting time \[ T=\sum_j \pi_j \mathbb E_i\tau_j \] (which is independent of \(i\); here, \(\mathbb E_i\) means the expectation with respect to the process starting from \(i\)). Then the following eigentime identity holds: \[ T=\sum_{n\geq1}\lambda_n^{-1}. \] The present paper generalizes the above eigentime identity for asymmetric (i.e., without reversibility assumption) finite Markov chains both in the ergodic case and the transient case, for continuous and discrete time.
scientific article; zbMATH DE number 5825457