The Invariant Measures of Markov Chains on Product Spaces and a Measurement of Dependence

For probability measures on product spaces, we define a notion of dependence among each coordinate. We study Markov chains on product spaces in which the time developement of each coordinate corresponds to the movement of a particle in an interacting particle system described by the Markov chain. If there is no interaction among these particles, the dependence of them decreases monotonically. We establish an inequality which states that if the interaction among each particle is small, then, in stationarity, the dependence among each particle is small.