Optimal stopping time and halting set for total variation distance

Agnes Coquio (1)

1 IF - Institut Fourier

An aperiodic and irreducible Markov chain on a finite state space converges to its stationary distribution. When convergence to equilibrium is measured by total variation distance, there exists an optimal coupling and a maximal coupling time. In this article, the maximal coupling time is compared to the hitting time of a specific state or set. Such sets, named halting sets, are studied in the case of symmetric birth-and-death chains and in some other examples. Some applications to the cutoff phenomenon are given. These results yield new methods to calculate cutoff times for some monotone birth-and death chains without the lazy hypothesis .

Mots clés: eigenvalues; passage time; total variation distance; stochastic monotonicity; Markov chains; birth-and-death chains

Domaines: Probabilités [math.PR]

Agnes Coquio. Optimal stopping time and halting set for total variation distance. 2015. ⟨hal-01136083⟩