Finite-time implications of relaxation times for stochastically monotone processes
A continuous-time Markov chain \((X_ t)\) which converges to a stationary distribution \(\pi\) often does so asymptotically at an exponential rate, that is, \(P(X_ t\in A)-\pi (A)\sim c_ A\exp (-\lambda t)\) as \(t\to \infty\). The quantity \(\tau\equiv 1/\lambda\) is then called the relaxation time of the chain. Motivated by the general question of what can be said rigorously about the finite-time behaviour of the chain given its stationary distribution \(\pi\) and its relaxation time \(\tau\), the author obtains a result for stochastically monotone chains for which the reversed chain is also stochastically monotone to the effect that \[ \sum_{i}\pi_ i\max_{j}| P_ i(X_ t\leq j)-\pi [0,j]| \leq 2(2+t/\tau)\exp (-t/t). \] Stochastic monotonicity is used here in the sense of monotonicity ``in space''.
scientific article; zbMATH DE number 4050685