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

Christopher Boucher, Richard S. Ellis and Bruce Turkington
The Annals of Probability
Vol. 27, No. 1 (Jan., 1999), pp. 297-324
Stable URL: http://www.jstor.org/stable/2652877
Page Count: 28
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Abstract

The main theorem is the large deviation principle for the doubly indexed sequence of random measures $W_{r,q}(dx x dy) \doteq \theta(dx) \otimes \sum^{2^r}_{k=1} 1_{D_r,k}(x)L_{q,k}(dy).$ Here θ is a probability measure on a Polish space X, {Dr,k, k = 1, ..., 2r} is a dyadic partition of X (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 y 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 two-dimensional turbulence as well as in a modification of that theory recently proposed by Turkington.

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