Spatializing Random Measures: Doubly Indexed Processes and the Large Deviation Principle

Here θ is a probability measure on a Polish space , Dr k k = 1 2r is a dyadic partition of (hence the use of 2r summands) satisfying θ Dr k = 1/2r and Lq 1 Lq 2 Lq 2r is an independent, identically distributed sequence of random probability measures on a Polish space such that Lq k q ∈ N satisfies the large deviation principle with a convex rate function. A number of related asymptotic results are also derived. The random measures Wr 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 twodimensional turbulence as well as in a modification of that theory recently proposed by Turkington.