Let @math be a discrete time Markov chain on a general state space. It is well-known that if @math is aperiodic and satisfies a drift and minorization condition, then it converges to its stationary distribution @math at an exponential rate. We consider the problem of computing upper bounds for the distance from stationarity in terms of the drift and minorization data. Baxendale showed that these bounds improve significantly if one assumes that @math is reversible with nonnegative eigenvalues (i.e. its transition kernel is a self-adjoint operator on @math with spectrum contained in @math ). We identify this phenomenon as a special case of a general principle: for a reversible chain with nonnegative eigenvalues, any strong random time gives direct control over the convergence rate. We formulate this principle precisely and deduce from it a stronger version of Baxendale's result. Our approach is fully quantitative and allows us to convert drift and minorization data into explicit convergence bounds. We show that these bounds are tighter than those of Rosenthal and Baxendale when applied to a well-studied example.
The most important feature of Theorem and its @math -norm version, Theorem , is that the exponential rate @math is the same as the decay rate in Theorem . As we will see in Section , this conclusion can only be drawn for reversible Markov chains with nonnegative eigenvalues and does not hold in general. Baxendale @cite_11 was the first to observe this consequence of reversibility. For a key argument, he credits a comment of Meyn on a previous draft of @cite_11 . In the case @math , Theorem is very similar to [Theorem 1.3] B05 : both theorems have the same hypotheses, and both prove @math -norm convergence of the chain with the same exponential rate @math .
@cite_11
2122337269 1547388036 2046434141 1593631888
We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are dierent from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition towards a “small set”. The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the first-entrance last-exit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory. In many applications of Markov chains, and especially in Markov chain Monte Carlo algorithms, the rate of convergence of the chain is of critical importance. Most techniques to establish such rates require bounds on the distribution of the random regeneration time T that can be constructed, via splitting techniques, at times of return to a "small set" C satisfying a minorisation condition P(x,Â·)[greater-or-equal, slanted][var epsilon][phi](Â·), x[set membership, variant]C. Typically, however, it is much easier to get bounds on the time [tau]C of return to the small set itself, usually based on a geometric drift function , where . We develop a new relationship between T and [tau]C, and this gives a bound on the tail of T, based on [var epsilon],[lambda] and b, which is a strict improvement on existing results. When evaluating rates of convergence we see that our bound usually gives considerable numerical improvement on previous expressions. Consider a graph, G, for which the vertices can have two modes, 0 or 1. Suppose that a particle moves around on G according to a discrete time Markov chain with the following rules. With (strictly positive) probabilities pm, pc and pr it moves to a randomly chosen neighbour, changes the mode of the vertex it is at or just stands still, respectively. We call such a random process a (pm, pc, pr)-lamplighter process on G. Assume that the process starts with the particle in a fixed position and with all vertices having mode 0. The convergence rate to stationarity in terms of the total variation norm is studied for the special cases with G = KN, the complete graph with N vertices, and when G = mod N. In the former case we prove that as N --> [infinity], ((2pc + pm) 4pcpm)N log N is a threshold for the convergence rate. In the latter case we show that the convergence rate is asymptotically determined by the cover time CN in that the total variation norm after aN2 steps is given by P(CN > aN2). The limit of this probability can in turn be calculated by considering a Brownian motion with two absorbing barriers. In particular, this means that there is no threshold for this case. Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, @math . The sequence is said to exhibit @math -cutoff if the convergence to stationarity in total variation distance is abrupt, w.r.t. this sequence of initial distributions. In this work we give a characterization of @math -cutoff for an arbitrary sequence of initial distributions @math (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. @math . Consider a Markov chain on @math whose stationary distribution in @math . Let @math be the expected hitting time of the worst set of size at least @math . It was recently proved by Peres and Sousi and independently by Oliveira that @math captures the order of the mixing time. In this work we further refine this connection and show that @math -cutoff can be characterized in terms of concentration of hitting times (starting from @math ) of sets which are worst in expectation w.r.t. @math . Conversely, we construct a counter-example which demonstrates that in general cutoff (as opposed to cutoff w.r.t. a certain sequence of initial distributions) cannot be characterized in this manner. Finally, we also prove that there exists an absolute constant @math such that for every Markov chain @math , for all @math , where @math is the inverse of the spectral gap of the chain.