We develop singular value techniques in the context of time inhomogeneous finite Markov chains with the goal of obtaining quantitative results concerning the asymptotic behavior of such chains. We introduce the notion of c-stability which can be viewed as a…

In this paper we investigate three types of convergence for geometrically ergodic Markov chains (MCs) with countable state space, which in general lead to different `rates of convergence'. For reversible Markov chains it is shown that these rates coincide. For…

Let $0<\alpha<1/2$. We show that that the mixing time of a continuous-time Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of the state space with stationary measure $\geq…

The best known lower and upper bounds on the mixing time for the random to-random insertions shuffle are $(1/2-o(1))n\log n$ and $(2+o(1))n\log n$. A long standing open problem is to prove that the mixing time exhibits a cutoff. In particular,…

Consider a deck of n cards. Let $p_1,p_2,\ldots,p_n$ be a probability vector and consider the mixing time of the card shuffle which at each step of time picks a position according to the pi 's and move the card in…