Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method

V. R. Fatalov

Izvestiya. Mathematics , Tome 75 (2011) no. 4, pp. 837-868.

Résumé

We prove results on exact asymptotics as $n\to\infty$ for the expectations $\mathsf{E}_a \exp\bigl\{-\theta\sum_{k=0}^{n-1} g(X_k)\bigr\}$ and probabilities $\mathsf{P}_a\bigl\{\frac{1}{n}\sum_{k=0}^{n-1}g(X_k)$, where $\{\xi_k\}_{k=1}^\infty $ is a sequence of independent identically Laplace-distributed random variables, $X_n=X_0+\sum_{k=1}^n \xi_k$, $n\geqslant 1$, is the corresponding random walk on $\mathbb{R}$, $g(x)$ is a positive continuous function satisfying certain conditions, and $d>0$, $\theta>0$, $a\in\mathbb{R}$ are fixed numbers. Our results are obtained using a new method which is developed in this paper: the Laplace method for the occupation time of discrete-time Markov chains. For $g(x)$ one can take $|x|^p$, $\log(|x|^p+1)$, $p>0$, $|x|\log(|x|+1)$, or $e^{\alpha |x|}-1$, $0\alpha1/2$, $x\in\mathbb{R}$, for example. We give a detailed treatment of the case when $g(x)=|x|$ using Bessel functions to make explicit calculations.

Détail

Keywords: large deviations, Laplace method, action functional, occupation time, Bessel function.
Mots-clés : Markov chains

Bibliographie

[1] M. D. Donsker, S. R. S. Varadhan, “Asymptotic evaluation of certain Markov process expectations for large time. III”, Comm. Pure Appl. Math., 29:4 (1976), 389–461 | DOI | MR | Zbl

[2] J.-D. Deuschel, D. W. Stroock, Large deviations, Pure Appl. Math., 137, Academic Press, Boston, MA, 1989 | MR | Zbl

[3] P. Dupuis, R. S. Ellis, A weak convergence approach to the theory of large deviations, Wiley Ser. Probab. Statist., Wiley, New York, 1997 | MR | Zbl

[4] V. I. Piterbarg, V. R. Fatalov, “The Laplace method for probability measures in Banach spaces”, Russian Math. Surveys, 50:6 (1995), 1151–1239 | DOI | MR | Zbl

[5] L. M. Wu, “Grandes déviations pour les processus de Markov essentiellement irréductibles. I. Temps discret”, C. R. Acad. Sci. Paris Sér. I Math., 312:8 (1991), 609–614 ; “II. Temps continu”, C. R. Acad. Sci. Paris Sér. I Math., 314:12 (1992), 941–946 ; “III. Quelques applications”, C. R. Acad. Sci. Paris Sér. I Math., 316:8 (1993), 853–858 | MR | Zbl | MR | Zbl | MR | Zbl

[6] L. Wu, “Moderate deviations of dependent random variables related to CLT”, Ann. Probab., 23:1 (1995), 420–445 | DOI | MR | Zbl

[7] L. Wu, “Essential spectral radius for Markov semigroups (I): discrete time case”, Probab. Theory Related Fields, 128:2 (2004), 255–321 | DOI | MR | Zbl

[8] A. de Acosta, “Moderate deviations for empirical measures of Markov chains: lower bounds”, Ann. Probab., 25:1 (1997), 259–284 | DOI | MR | Zbl

[9] A. de Acosta, X. Chen, “Moderate deviations for empirical measures of Markov chains: upper bounds”, J. Theoret. Probab., 11:4 (1998), 1075–1110 | DOI | MR | Zbl

[10] J. L. Jensen, “Saddlepoint expansions for sums of Markov dependent variables on a continuous state space”, Probab. Theory Related Fields, 89:2 (1991), 181–199 | DOI | MR | Zbl

[11] G. Lerman, Z. Schuss, “Asymptotic theory of large deviations for Markov chains”, SIAM J. Appl. Math., 58:6 (1998), 1862–1877 | DOI | MR | Zbl

[12] H. Djellout, A. Guillin, “Principe de déviations modérées pour le processus empirique fonctionnel d'une chaîne de Markov”, C. R. Acad. Sci. Paris Sér. I Math., 330:5 (2000), 377–380 | DOI | MR | Zbl

[13] A. A. Borovkov, K. A. Borovkov, “On probabilities of large deviations for random walks. I. Regularly varying distribution tails”, Theory Probab. Appl., 46:2 (2002), 193–213 | DOI | MR | Zbl

[14] A. A. Borovkov, A. A. Mogul'skii, “Large deviations for Markov chains in the positive quadrant”, Russian Math. Surveys, 56:5 (2001), 803–916 | DOI | MR | Zbl

[15] Sh. Kusuoka, Y. Tamura, “Precise estimate for large deviation of Donsker–Varadhan type”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38:3 (1991), 533–565 | MR | Zbl

[16] E. Bolthausen, J.-D. Deuschel, Y. Tamura, “Laplace approximations for large deviations of nonreversible Markov processes. The nondegenerate case”, Ann. Probab., 23:1 (1995), 236–267 | DOI | MR | Zbl

[17] S. Liang, J. Liu, “On the precise Laplace approximation for large deviations of Markov chain. The nondegenerate case”, J. Math. Sci. Univ. Tokyo, 10:2 (2003), 421–454 | MR | Zbl

[18] V. R. Fatalov, “The Laplace method for small deviations of Gaussian processes of Wiener type”, Sb. Math., 196:4 (2005), 595–620 | DOI | MR | Zbl

[19] V. R. Fatalov, “Occupation times and exact asymptotics of small deviations of Bessel processes for $L^p$-norms with $p>0$”, Izv. Math., 71:4 (2007), 721–752 | DOI | MR | Zbl

[20] V. R. Fatalov, “Occupation time and exact asymptotics of distributions of $L^p$-functionals of the Ornstein–Uhlenbeck processes, $p>0$”, Probl. Inf. Transm., 53:1 (2009), 13–36 | DOI | Zbl

[21] S. Albeverio, V. Fatalov, V. Piterbarg, “Asymptotic behavior of the sample mean of a function of the Wiener process and the MacDonald function”, J. Math. Sci. Univ. Tokyo, 16:1 (2009), 55–93 | MR | Zbl

[22] E. Nummelin, General irreducible Markov chains and nonnegative operators, Cambridge Tracts in Math., 83, Cambridge Univ. Press, Cambridge, 1984 | MR | MR | Zbl

[23] D. Revyuz, Tsepi Markova, RFFI, M., 1997

[24] V. Koroliouk, N. Portenko, A. Skorokhod, A. Tourbine, Aide-mémoire de théorie des probabilités et de statistique mathématique, Mir, Moscow, 1983 | MR | MR | Zbl | Zbl

[25] F. Spitzer, Principles of random walks, Springer-Verlag, New York–Heidelberg, 1976 | MR | Zbl | Zbl

[26] A. N. Borodin, I. A. Ibragimov, “Limit theorems for functionals of random walks”, Proc. Steklov Inst. Math., 195 (1995), 1–259 | MR | Zbl | Zbl

[27] V. V. Petrov, Sums of independent random variables, Springer-Verlag, Berlin–Heidelberg–New York, 1975 | MR | MR | Zbl | Zbl

[28] F. D. Gakhov, Yu. I. Cherskii, Uravneniya tipa svertki, Nauka, M., 1978 | MR | Zbl

[29] N. C. Jain, “An introduction to large deviations”, Probability in Banach spaces (Medford, MA, 1984), v. V, Lect. Notes Math., 1153, Springer-Verlag, Berlin, 1985, 273–296 | DOI | MR | Zbl

[30] F. W. J. Olver, Asymptotics and special functions, Academic Press, New York–London, 1974 | MR | MR | Zbl | Zbl

[31] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, U.S. Government Printing Office, Washington, 1964 | MR | MR | Zbl | Zbl

[32] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, vols. I, II, McGraw-Hill, New York–Toronto–London, 1953 | MR | MR | MR | Zbl | Zbl

[33] M. Kac, “On the average of a certain Wiener functional and a related limit theorem in calculus of probability”, Trans. Amer. Math. Soc., 59:3 (1946), 401–414 | DOI | MR | Zbl

[34] V. R. Fatalov, “Large deviations for distributions of sums of random variables: Markov chain method”, Probl. Inf. Transm., 46:2 (2010), 160–183 | DOI | MR

[35] S. R. S. Varadhan, “Asymptotic probabilities and differential equations”, Comm. Pure Appl. Math., 19 (1966), 261–286 | DOI | MR | Zbl

[36] M. M. Vainberg, Variational method and method of monotone operators in the theory of nonlinear equations, Halsted Press, New York–Toronto, 1973 | MR | MR | Zbl | Zbl

[37] A. D. Polyanin, A. V. Manzhirov, Handbook of integral equations, CRC Press, Boca Raton, FL, 1998 | MR | Zbl | Zbl

[38] V. M. Alekseev, V. M. Tikhomirov, S. V. Fomin, Optimal control, Contemp. Soviet Math., Consultants Bureau, New York, 1987 | MR | MR | Zbl | Zbl

[39] E. Bolthausen, “Laplace approximations for sums of independent random vectors”, Probab. Theory Relat. Fields, 72:2 (1986), 305–318 | DOI | MR | Zbl

[40] V. R. Fatalov, “The Laplace method for computing exact asymptotics of distributions of integral statistics”, Math. Methods Statist., 8:4 (1999), 510–535 | MR | Zbl

[41] B. A. Sevastyanov, Vetvyaschiesya protsessy, Nauka, M., 1971 | MR | Zbl

[42] L. Lei, “Large deviations of kernel density estimator in $L^1(\mathbb{R}^d)$ for reversible Markov processes”, Bernoulli, 12:1 (2006), 65–83 | MR | Zbl

[43] J. Kerstan, K. Matthes, J. Mecke, Unbegrenzt teilbare Punktprozesse, Akademie-Verlag, Berlin, 1974 | MR | MR | Zbl | Zbl

[44] L. V. Kantorovich, G. P. Akilov, Functional analysis, Pergamon Press, Oxford, 1982 | MR | MR | Zbl | Zbl