Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Precise large deviation results for products of random matrices

Dariusz Buraczewski and Sebastian Mentemeier

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of large numbers for the norm of a vector on which the matrices act. We prove corresponding precise large deviation results, generalizing the Bahadur–Rao theorem to this situation. Therefore, we obtain a third-order Edgeworth expansion for the cumulative distribution function of the vector norm. This result in turn relies on an application of the Nagaev–Guivarch method. Our result is then used to study matrix recursions, arising e.g. in financial time series, and to provide precise large deviation estimates there.

Résumé

Le théorème de Furstenberg et Kesten établit une loi forte des grands nombres pour la norme d’un produit de matrices aléatoires. Cela peut être étendu sous des hypothèses variées, dans le cas des matrices positives ou inversibles, à une loi des grand nombres pour la norme d’un vecteur sur lequel les matrices agissent. Dans ce cadre, nous prouvons des résultats de grandes déviations précis, en généralisant le théorème de Bahadur–Rao à cette situation. Ainsi, nous obtenons une expansion de Edgeworth au troisième ordre pour la fonction de répartition de la norme du vecteur. Ce résultat se base sur une application de la méthode de Nagaev–Guivarch. Notre résultat est utilisé ensuite pour étudier des récurrences matricielles, qui apparaissent par exemple dans les séries temporelles en finance, et pour donner des estimations précises de grandes déviations.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1474-1513.

Dates
Received: 26 May 2014
Revised: 16 April 2015
Accepted: 16 April 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723527

Digital Object Identifier
doi:10.1214/15-AIHP684

Mathematical Reviews number (MathSciNet)
MR3531716

Zentralblatt MATH identifier
1357.60028

Subjects
Primary: 60F10: Large deviations
Secondary: 60H25: Random operators and equations [See also 47B80]

Keywords
Products of random matrices Limit theorems Large deviations Random difference equations Edgeworth expansion Fourier techniques Markov chains with general state space Markov random walks Heavy tails

Citation

Buraczewski, Dariusz; Mentemeier, Sebastian. Precise large deviation results for products of random matrices. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1474--1513. doi:10.1214/15-AIHP684. https://projecteuclid.org/euclid.aihp/1469723527


Export citation

References

  • [1] G. Alsmeyer and S. Mentemeier. Tail behaviour of stationary solutions of random difference equations: The case of regular matrices. J. Difference Equ. Appl. 18 (8) (2012) 1305–1332.
  • [2] B. Basrak, R. A. Davis and T. Mikosch. Regular variation of GARCH processes. Stochastic Process. Appl. 99 (1) (2002) 95–115.
  • [3] A. Behme and A. Lindner. Multivariate generalized Ornstein–Uhlenbeck processes. Stochastic Process. Appl. 122 (4) (2012) 1487–1518.
  • [4] P. Bougerol and J. Lacroix. Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser, Boston, 1985.
  • [5] P. Bougerol and N. Picard. Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (4) (1992) 1714–1730.
  • [6] D. Buraczewski and S. Mentemeier. Precise tail asymptotics for attracting fixed points of multivariate smoothing transformations. Preprint, 2015. Available at arXiv:1502.02397.
  • [7] D. Buraczewski, E. Damek and Y. Guivarc’h. Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Related Fields 148 (3–4) (2010) 333–402.
  • [8] D. Buraczewski, E. Damek, Y. Guivarc’h, A. Hulanicki and R. Urban. Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Related Fields 145 (3–4) (2009) 385–420.
  • [9] D. Buraczewski, E. Damek, Y. Guivarc’h and S. Mentemeier. On multidimensional Mandelbrot’s cascades. J. Difference Equ. Appl. 20 (11) (2014) 1523–1567.
  • [10] D. Buraczewski, E. Damek, S. Mentemeier and M. Mirek. Heavy tailed solutions of multivariate smoothing transforms. Stochastic Process. Appl. 123 (6) (2013) 1947–1986.
  • [11] D. Buraczewski, E. Damek and J. Zienkiewicz. Precise tail asymptotics of fixed points of the smoothing transform with general weights. Bernoulli 21 (1) (2015) 489–504.
  • [12] H. Cohn, O. Nerman and M. Peligrad. Weak ergodicity and products of random matrices. J. Theoret. Probab. 6 (2) (1993) 389–405.
  • [13] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Applications of Mathematics (New York) 38. Springer, New York, 1998.
  • [14] W. Feller. An Introduction to Probability Theory and Its Applications. Vol. II, 2nd edition. Wiley, New York, 1971.
  • [15] H. Furstenberg and H. Kesten. Products of random matrices. Ann. Math. Statist. 31 (1960) 457–469.
  • [16] Y. Guivarc’h. Spectral gap properties and limit theorems for some random walks and dynamical systems. In Hyperbolic Dynamics, Fluctuations and Large Deviations 279–310. D. Dolgopyat, Y. Pesin, M. Pollicott and L. Stoyanov (Eds). Proceedings of Symposia in Pure Mathematics 89, 2015.
  • [17] Y. Guivarc’h and É. Le Page. Spectral gap properties and asymptotics of stationary measures for affine random walks. Ann. Inst. Henri Poincaré Probab. Stat. To appear.
  • [18] Y. Guivarc’h and R. Urban. Semigroup actions on tori and stationary measures on projective spaces. Studia Math. 171 (1) (2005) 33–66.
  • [19] H. Hennion. Limit theorems for products of positive random matrices. Ann. Probab. 25 (4) (1997) 1545–1587.
  • [20] H. Hennion and L. Hervé. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics 1766. Springer, Berlin, 2001.
  • [21] L. Hervé and F. Pène. The Nagaev–Guivarc’h method via the Keller–Liverani theorem. Bull. Soc. Math. France 138 (3) (2010) 415–489.
  • [22] M. Iltis. Sharp asymptotics of large deviations for general state-space Markov-additive chains in $\mathbf{R}^{d}$. Statist. Probab. Lett. 47 (4) (2000) 365–380.
  • [23] H. Kesten. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973) 207–248.
  • [24] J. F. C. Kingman. Subadditive ergodic theory. Ann. Probab. 1 (1973) 883–909.
  • [25] C. Klüppelberg and S. Pergamenchtchikov. The tail of the stationary distribution of a random coefficient $\operatorname{AR}(q)$ model. Ann. Appl. Probab. 14 (2) (2004) 971–1005.
  • [26] I. Kontoyiannis and S. P. Meyn. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (1) (2003) 304–362.
  • [27] É. Le Page. Théorèmes limites pour les produits de matrices aléatoires. In Probability Measures on Groups (Oberwolfach, 1981) 258–303. Lecture Notes in Mathematics 928. Springer, Berlin, 1982.
  • [28] É. Le Page. Théorèmes de renouvellement pour les produits de matrices aléatoires. In Séminaires de Probabilités Rennes 1983 1–116. Publication des Séminaires de Mathématiques, Univ. Rennes I.
  • [29] S. Mentemeier. On multivariate stochastic fixed point equations: The smoothing transform and random difference equations. Ph.D. thesis, Westfälische Wilhelms-Universität Münster, 2013.
  • [30] M. Mirek. On fixed points of a generalized multidimensional affine recursion. Probab. Theory Related Fields 156 (3–4) (2013) 665–705.
  • [31] P. Ney and E. Nummelin. Markov additive processes II. Large deviations. Ann. Probab. 15 (2) (1987) 593–609.
  • [32] H. M. Wang. A note on multitype branching process with bounded immigration in random environment. Acta Math. Sin. (Engl. Ser.) 29 (6) (2013) 1095–1110.
  • [33] K. Yosida. Functional Analysis, 6th edition. Grundlehren der Mathematischen Wissenschaften 123. Springer, Berlin, 1980.