The Annals of Probability

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

Christopher Boucher, Richard S. Ellis, and Bruce Turkington

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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 $L_{q,1}L_{q,2},\ldotsL_{q,2_r}$ is an independent, identically distributed sequesnce of random probability measures on a Ploish space$ \mathscr{Y}$ such that ${L_{q,k}q\in \mathsbb{N}}$ 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.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 297-324.

First available in Project Euclid: 29 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations
Secondary: 82B99: None of the above, but in this section

Large deviation principle doubly indexed processes random measures Sanov’s theorem turbulence


Boucher, Christopher; Ellis, Richard S.; Turkington, Bruce. Spatializing Random Measures: Doubly Indexed Processes and the Large Deviation Principle. Ann. Probab. 27 (1999), no. 1, 297--324. doi:10.1214/aop/1022677264.

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