Large and Moderate Deviations for Projective Systems and Projective Limits

Tryfon Daras

Department of Sciences, Technical University of Crete, Chania, Greece.

DOI: 10.4236/am.2012.312A282                

Abstract

One of the active fields in applied probability, the last two decades, is that of large deviations theory i.e. the one dealing with the (asymptotic) computation of probabilities of rare events which are exponentially small as a function of some parameter e.g. the amplitude of the noise perturbing a dynamical system. Basic ideas of the theory can be tracked back to Laplace, the first rigorous results are due to Cramer although a clear definition was introduced by Varadhan in 1966. Large deviations estimates have been proved to be the crucial tool in studying problems in Statistics, Physics (Thermodynamics and Statistical Mechanics), Finance (Monte-Carlo methods, option pricing, long term portfolio investment) and in Applied probability (queuing theory). The aim of this work is to describe one of the (recent) methods of proving large deviations results, namely that of projective systems. We compare the method with the one of projective limits and show the advantages of the first. These advantages are due to the fact that: 1) the arguments are direct and the proofs of the basic results of the theory are much easier and simpler; 2) we are able to extend most of these results using suitable projective systems. We apply the method in the case of a) sequences of i.i.d. r.v.’s and b) sequences of exchangeable r.v.’s. All the results are being proved in a simple “unified” way.

Keywords

Large Deviations; Projective Systems; Projective Limits; Moderate Deviations

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Dembo and O. Zeitouni, “Large Deviations Techniques and Applications,” Jones and Bartlett, Boston, 1993.
[2]	D. W. Strook, “An Introduction to the Theory of Large Deviations,” Springer-Verlag, New York, 1984. doi:10.1007/978-1-4613-8514-1
[3]	A. de Acosta, “Exponential Tightness and Projective Systems in Large Deviation Theory,” In: D. Pollard, E. Togersen and G. Yang, Eds., Festschrift for Lucien Le Cam, Springer, New York, 1997, pp. 143-156,
[4]	J. A. Bucklew, “Large Deviation Techniques in Decision, Simulation and Estimation,” John Wiley & Sons, New York, 1990.
[5]	P. Dupuis and R. Ellis, “A Weak Convergence Approach to the Large Deviations,” Wiley Series in Probability, New York, 1997.
[6]	T. Daras, “Large and Moderate Deviations for the Empirical Measures of an Exchangeable Sequence,” Statistics & Probability Letters, Vol. 36, No. 1, 1997, pp. 91-100. doi:10.1016/S0167-7152(97)00052-7
[7]	A. de Acosta, “On Large Deviations of Empirical Measures in the Τ-Topology,” Journal of Applied Probability, Vol. 31, 1994, pp. 41-47.
[8]	D. J. Aldous, “Exchangeability and Related Topics. Ecole d’ Ete de Probabilites de Saint-Flour XIII 1983,” Lecture Notes in Mathematics 117, Springer, New York, 1985.
[9]	I. H. Dinwoodie and S. L. Zabell, “Large Deviations for Exchangeable Random Vectors,” The Annals of Prob- ability, Vol. 20, No. 3, 1992, pp. 1147-1166. doi:10.1214/aop/1176989683
[10]	T. Daras, “Trajectories of Exchangeable Sequences: Large and Moderate Deviations Results,” Statistics & Probability Letters, Vol. 39, No. 4, 1998, pp. 289-304.
[11]	A. de Acosta, “Projective Systems in Large Deviation Theory II: Some Applications,” Probability in Banach Spaces 9, Vol. 35, 1994, pp. 241-250.