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

Abstract

The main theorem is the large deviation principle for the doubly indexed sequence of random measures $$W_{r,q}(dx \times dy) \doteq \theta(dx) \otimes \textstyle\sum\limits_{k=1}^{2^r} 1_{D_{r,k}}(x) L_{q,k} (dy).$$ Here $\theta$ is a probability measure on a Polish space $\mathscr{X},\{D_{r,k}k=1,\ldots,2^r\}$ is a dyadic partition of $\mathscr{X}$ (hence the use of $2^r$ summands) satisfying $\theta\{D_{r,k}\}= 1/2^r$ and $L_{q,1}L_{q,2},\ldots , L_{q,2_r}$ is an independent, identically distributed sequesnce of random probability measures on a Ploish space $ \mathscr{Y}$ such that $\{L_{q,k}q\in \mathbb{N}\}$ satisfies the large deviation principle with a convex rate function. A number of related asymptotic results are also derived.

The random measures $W_{ r,q}$ 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.