The Annals of Applied Probability

Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard

Françoise Pène

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Abstract

We show how Rio’s method [Probab. Theory Related Fields 104 (1996) 255–282] can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt{n}}}$ in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 4 (2005), 2331-2392.

Dates
First available in Project Euclid: 7 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1133965765

Digital Object Identifier
doi:10.1214/105051605000000476

Mathematical Reviews number (MathSciNet)
MR2187297

Zentralblatt MATH identifier
1097.37030

Subjects
Primary: 37D50: Hyperbolic systems with singularities (billiards, etc.) 60F05: Central limit and other weak theorems

Keywords
Multidimensional central limit theorem Kantorovich metric Prokhorov metric rate of convergence

Citation

Pène, Françoise. Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard. Ann. Appl. Probab. 15 (2005), no. 4, 2331--2392. doi:10.1214/105051605000000476. https://projecteuclid.org/euclid.aoap/1133965765


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References

  • Bergström, H. (1944). On the central limit theorem. Skand. Aktuarie Tidskr. 27 139–153.
  • Bergström, H. (1945). On the central limit theorem in the space $R_k$, $k>1$. Skand. Aktuarie Tidskr. 28 106–127.
  • Berry, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49 122–136.
  • Bhattacharya, R. N. (1970). Rates of weak convergence for the multidimensional central limit theorem. Theory Probab. Appl. 15 68–86.\goodbreak
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Boatto, S. and Golse, F. (2002). Diffusion approximation of a Knudsen gas model: Dependence of the diffusion constant upon a boundary condition. Asymptotic Anal. 31 93–111.
  • Bolthausen, E. (1982). Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 672–688.
  • Bunimovich, L. A., Chernov, N. I. and Sinai, Ya. G. (1990). Markov partitions for two-dimensional hyperbolic billiards. Russian Math. Surveys 45 105–152.
  • Bunimovich, L. A., Chernov, N. I. and Sinai, Ya. G. (1991). Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys 46 47–106.
  • Bunimovich, L. A. and Sinai, Ya. G. (1980). Markov partitions for dispersed billiards. Comm. Math. Phys. 78 247–280.
  • Bunimovich, L. A. and Sinai, Ya. G. (1981). Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 479–497.
  • Chernov, N. I. (1999). Decay of correlation and dispersing billiards. J. Statist. Phys. 94 513–556.
  • Conze, J.-P. and Le Borgne, S. (2001). Méthode de martingales et flot géodésique sur une surface de courbure négative constante. Ergodic Theory Dynam. Systems 21 421–441.
  • Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth and Brooks Cole, Pacific Grove, CA.
  • Esseen, C.-G. (1945). Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law. Acta Math. 77 1–125.
  • Feller, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • Gordin, M. I. (1969). The central limit theorem for stationary processes. Soviet Math. Dokl. 10 1174–1176. [Translation from Dokl. Akad. Nauk SSSR 188 (1969) 739–741.]
  • Guivarc'h, Y. and Hardy, J. (1988). Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov. Ann. Inst. H. Poincaré Probab. Statist. 24 73–98.
  • Guivarc'h, Y. and Raugi, A. (1986). Products of random matrices: Convergence theorems. Contemp. Math. 50 31–54.
  • Hennion, H. and Hervé, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, New York.
  • Ionescu Tulcea, C. (1949). Mesures dans les espaces produits. Atti Acad. Naz. Lincei Rend. 7.
  • Jan, C. (2000). Rates of convergence in the CLT for Markov chains and some dynamical systems processes. C. R. Acad. Sci. Paris Sér. I Math. 331 395–398.
  • Jan, C. (2001). Vitesse de convergence dans le TCL pour des processus associés à des systèmes dynamiques et aux produits de matrices aléatoires. Ph.D. thesis, Univ. Rennes 1.
  • Krámli, A., Simányi, N. and Szász, D. (1989). Dispersing billiards without focal points on surfaces are ergodic. Comm. Math. Phys. 125 439–457.
  • Le Borgne, S. and Pène, F. (2003). Vitesse dans le théorème limite central pour certains procesus stationnaires fortement décorrélés. arXiv:math.PR/0306083.
  • Le Borgne, S. and Pène, F. (2005). Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques. Bull. Soc. Math. France. To appear.
  • Le Jan, Y. (1994). The central limit theorem for the geodesic flow on noncompact manifolds of constant negative curvature. Duke Math. J. 74 159–175.
  • Mano, P. (1988). Ph.D. thesis, Univ. Paris VI.
  • Nagaev, S. V. (1957). Some limit theorems for stationary Markov chains. Theory Probab. Appl. 11 378–406.
  • Nagaev, S. V. (1961). More exact statements of limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6 62–81.
  • Nagaev, S. V. (1965). Some limit theorems for large deviations. Theory Probab. Appl. 10 214–235.
  • Neveu, J. (1970). Bases Mathématiques du Calcul des Probabilités. Masson and Cie, Paris.
  • Pène, F. (2001). Rates of convergence in the CLT for two-dimensional dispersive billiards. Comm. Math. Phys.
  • Pène, F. (2004). Multiple decorrelation and rate of convergence in multidimensional limit theorems for the Prokhorov metric. Ann. Probab. 32 2477–2525.
  • Ranga Rao, R. (1961). On the central limit theorem in $R_k$. Bull. Amer. Math. Soc. 67 359–361.
  • Ratner, M. (1973). The central limit theorem for geodesic flows on $n$-dimensional manifolds of negative curvature. Israel J. Math. 16 181–197.
  • Rio, E. (1996). Sur le théorème de Berry–Esseen pour les suites faiblement dépendantes. Probab. Theory Related Fields 104 255–282.
  • Rotar, V. I. (1970). A non-uniform estimate for the convergence in the multi-dimensional central limit theorem. Teor. Veroyatnost. i Primenen. 15 647–665.
  • Sinai, Ya. G. (1960). The central limit theorem for geodesic flows on manifolds of constant negative curvature. Dokl. Akad. Nauk SSSR 133 1303–1306. [Translation in Soviet Math. Dokl. 1 (1960) 983–987.]
  • Sinai, Ya. G. (1970). Dynamical systems with elastic reflections. Russian Math. Surveys 25 137–189.
  • Sunklodas, Ĭ. (1982). Distance in the $L_1$ metric of the distribution of the sum of weakly dependent random variables from the normal distribution function. Litovsk. Mat. Sb. 22 171–188.
  • Sweeting, T. J. (1977). Speeds of convergence for the multidimensional central limit theorem. Ann. Probab. 5 28–41.
  • Young, L.-S. (1998). Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 585–650.
  • Yurinskii, V. V. (1975). A smoothing inequality for estimates of the Levy–Prokhorov distance. Theory Probab. Appl. 20 1–10. [Translation from Teor. Veroyatnost. i Primenen. 20 (1975) 3–12.]