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.
In the case @math , the decay rate @math of the law of @math was identified by Roberts and Tweedie @cite_18 (who use the notation @math ). Theorem 4.1(i) of @cite_18 is equivalent to a bound of the form [ (T > t) ( const ) (V)^r t ^t. ] Theorem slightly improves this result by removing the factor of @math and generalizing to the case @math .
@cite_18
1547388036
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.