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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Abstract

The main theorem is the large deviation principle for the doubly indexed sequence of random measures

\[ W_{r,q}(dx \times dy) \doteq \theta(dx) \otimes \textstyle\sum\limits_{k=1}^{2^r} 1_{D_{r,k}}(x) L_{q,k} (dy) \]

Abstract

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

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

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677264

Digital Object Identifier
doi:10.1214/aop/1022677264

Mathematical Reviews number (MathSciNet)
MR1681102

Zentralblatt MATH identifier
0942.60016

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1022677264


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References

  • [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [2] Boucher, C., Ellis, R. S. and Turkington, B. (1998). Derivation of maximum entropy principles in two-dimensional turbulence via large deviations. Preprint.
  • [3] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Their Applications. Jones and Bartlett, Boston.
  • [4] Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston.
  • [5] Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA.
  • [6] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
  • [7] Eisele, T. and Ellis, R. S. (1983). Symmetry breaking and random waves for magnetic systems on a circle.Wahrsch. Verw. Gebiete 63 279-348.
  • [8] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York.
  • [9] Ellis, R. S. (1988). Large deviations for the empirical measure of a Markov chain with an application to the multivariate empirical measure. Ann. Probab. 16 1496-1508.
  • [10] Ellis, R. S. (1995). An overview of the theory of large deviations and applications to statistical mechanics. Scand. Actuar. J. 1 97-142.
  • [11] Ellis, R. S. and Wyner, A. D. (1989). Uniform large deviation property of the empirical process of a Markov chain. Ann. Probab. 17 1147-1151.
  • [12] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [13] F ¨ollmer, H. and Orey, S. (1987). Large deviations for the empirical field of a Gibbs measure. Ann. Probab. 16 961-977.
  • [14] Lynch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Probab. 15 610-627.
  • [15] Michel, J. and Robert, R. (1994). Large deviations for Young measures and statistical mechanics of infinite dimensional dynamical systems with conservation law. Comm. Math. Phys. 159 195-215.
  • [16] Miller, J. (1990). Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett. 65 2137-2140.
  • [17] Olla, S. (1988). Large deviations for Gibbs random fields. Probab. Theory Related Fields 77 343-359.
  • [18] Robert, R. (1989). Concentration et entropie pour les mesures d'Young." C. R. Acad. Sci. Paris S´er. I 309 757-760.
  • [19] Robert, R. (1991). A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Statist. Phys. 65 531-553.
  • [20] Shiryaev, A. N. (1996). Probability, 2nd ed. Springer, New York.
  • [21] Stroock, D. W. (1984). An Introduction to the Theory of Large Deviations. Springer, New York.
  • [22] Turkington, B. (1999). Statistical equilibrium measures and coherent states in twodimensional turbulence. Comm. Pure Appl. Math. To appear.