Spatializing random measures: Doubly indexed processes and the large deviation principle

The main theorem is the large deviation principle for the doubly indexed sequence of random measures [GRAPHICS] Here theta is a probability measure on a Polish space K, {D-r, k, k = 1,...,2(r)} is a dyadic partition of K (hence the use of 2(r) summands) satisfying theta{D-r,D-k} = 1/2(r) and L-q,(1), L-q,(2),...,L-q, (2r) is an independent, identically distributed sequence of random probability measures on a Polish space Y such that {L-q,L-k, q is an element of N} satisfies the large deviation principle with a convex rate function. A number of related asymptotic results are also derived. The random measures W-r,W-q have important applications to the statistical mechanics of turbulence. In a companion paper, the large deviation principle presented here is used to give a rigorous derivation of maximum entropy principles arising in the well-known Miller-Robert theory of two-dimensional turbulence as well as in a modification of that theory recently proposed by Turkington.