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January 1999 Spatializing Random Measures: Doubly Indexed Processes and the Large Deviation Principle
Christopher Boucher, Richard S. Ellis, Bruce Turkington
Ann. Probab. 27(1): 297-324 (January 1999). DOI: 10.1214/aop/1022677264


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.


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Christopher Boucher. Richard S. Ellis. Bruce Turkington. "Spatializing Random Measures: Doubly Indexed Processes and the Large Deviation Principle." Ann. Probab. 27 (1) 297 - 324, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0942.60016
MathSciNet: MR1681102
Digital Object Identifier: 10.1214/aop/1022677264

Primary: 60F10
Secondary: 82B99

Keywords: doubly indexed processes , large deviation principle , Random measures , Sanov’s theorem , turbulence

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 1 • January 1999
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