Consistent estimation of joint distributions for sufficiently mixing random fields

Abstract

The joint distribution of a d-dimensional random field restricted to a box of size k can be estimated by looking at a realization in a box of size $n \gg k$ and computing the empirical distribution. This is done by sliding a box of size k around in the box of size n and computing frequencies. We show that when $k = k(n)$ grows as a function of n, then the total variation distance between this empirical distribution and the true distribution goes to 0 a.s. as $n \to \infty$ provided $k(n)^d \leq (\log n^d)/(H + \varepsilon)$ (where H is the entropy of the random field) and providing the random field satisfies a condition called quite weak Bernoulli with exponential rate. This class of processes, studied previously, includes the plus state for the Ising model at a variety of parameter values and certain measures of maximal entropy for certain subshifts of finite type. Marton and Shields have proved such results in one dimension and this paper is an attempt to extend their results to some extent to higher dimensions.