We introduce a new partial order on the class of stochastically monotone
Markov kernels having a given stationary distribution Ο on a given finite
partially ordered state space X. When Kβͺ―L in this partial
order we say that K and L satisfy a comparison inequality. We establish
that if K1β,β¦,Ktβ and L1β,β¦,Ltβ are reversible and Ksββͺ―Lsβ for s=1,β¦,t, then K1ββ―Ktββͺ―L1ββ―Ltβ. In
particular, in the time-homogeneous case we have Ktβͺ―Lt for every t
if K and L are reversible and Kβͺ―L, and using this we show that
(for suitable common initial distributions) the Markov chain Y with kernel
K mixes faster than the chain Z with kernel L, in the strong sense that
at every time t the discrepancy - measured by total variation distance or
separation or L2-distance - between the law of Ytβ and Ο is smaller
than that between the law of Ztβ and Ο. Using comparison inequalities
together with specialized arguments to remove the stochastic monotonicity
restriction, we answer a question of Persi Diaconis by showing that, among all
symmetric birth-and-death kernels on the path X={0,β¦,n}, the
one (we call it the uniform chain) that produces fastest convergence from
initial state 0 to the uniform distribution has transition probability 1/2 in
each direction along each edge of the path, with holding probability 1/2 at
each endpoint.Comment: Published in at http://dx.doi.org/10.1214/12-AAP886 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org