Abstract
In this chapter we introduce the main concepts of large deviations theory. We state some of the main theorems with several examples, from Cramér theorem for the sum of independent random variables, to Freidlin–Wentzell theory of random perturbation of dynamical systems.
Keywords
- Central Limit Theorem
- Relative Entropy
- Empirical Measure
- Large Deviation Principle
- Contraction Principle
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Azencott, R.: Grandes déviations et Applications. In: Ecole d’Eté de Probabilities de Saint-Flour VIII-1978. Lecture Notes in Math., vol. 774, pp. 1–176. Springer, New York (1980)
Baldi, P.: Large deviations and stochastic homogenization. Ann. Mat. Pura Appl. 151, 161–177 (1988)
Bolthausen, E.: Laplace approximations for sums of independent random vectors. Probab. Theor. Relat. Field 71(2), 167–206 (1987)
Boltzmann, L.: Über die beziehung zwischen dem zweiten hauptsatze der mechanischen wärmetheoreie un der Wahrscheinlichkeitrechnung respektive den sätzen über das wärmegleichgewicht (On the relationship between the second law of the mechanical theory of heat and the probability calculus). Wiener Berichte 2(76), 373–435 (1877)
Bryc, W.: A remark on the connection between the large deviation principle and the central limit theorem. Stat. Probab. Lett. 18(4), 253–256 (1993)
Cramér, H.: Sur un nouveau théorème limite dans la théorie des probabilités. Colloque consacré à la théorie des probabilités, Hermann. 3, 2–29 (1938)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)
den Hollander, F.: Large Deviations. Fields Institute Monograph. American Mathematical Society, Providence (2000)
Deuschel, J.D., Stroock, D.W.: Large Deviations. Academic, New York (1989)
Ellis, R.S.: Large deviations for a general class of random vectors. Ann. Probab. 12, 1–12 (1984)
Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)
Ethier, S.N., Kurtz, T.G.: Markov Processes. Wiley, New York (1986)
Freidlin, M., Wentzell, A.D.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York (1998)
Gärtner, J.: On large deviations from the invariant measure. Theor. Probab. Appl. 22, 24–39, (1977)
Pakdaman, K., Thieullen, M., Wainrib, G.: Diffusion approximation of birth-death processes: comparison in terms of large deviations and exit point. Stat. Probab. Lett. 80(13–14), 1121–1127 (2010)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1997)
Sanov, I.N.: On the probability of large deviations of random variables. Selected translations in Math. Stat. Probab. I 213–244 (1961)
Schilder, M.: Some asymptotic formulae for Wiener integrals. Trans. Am. Math. Soc. 125, 63–85 (1966)
Touchette, H.: The large deviations approach to statistical mechanics. Phys. Rep. 478, 1–69 (2009)
Varadhan, S.R.S.: Large Deviations and Applications. SIAM, Philadelphia (1984)
Varadhan, S.R.S.: Lectures on hydrodynamic scaling. In: Hydrodynamic Limits and Related Topics. Fields Institute Communication, vol. 27, pp. 3–42. American Mathematical Society, Providence (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wainrib, G. (2013). A Brief Introduction to Large Deviations Theory. In: Bachar, M., Batzel, J., Ditlevsen, S. (eds) Stochastic Biomathematical Models. Lecture Notes in Mathematics(), vol 2058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32157-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-32157-3_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32156-6
Online ISBN: 978-3-642-32157-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)