A central limit theorem for normalized products of random matrices
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A CENTRAL LIMIT THEOREM FOR NORMALIZED PRODUCTS OF RANDOM MATRICES ´ndez-Herna ´ndez2 Rolando Cavazos-Cadena1 and Daniel Herna [Communicated by Istv´ an Berkes] 1
Departamento de Estad´ıstica y C´ alculo, Universidad Aut´ onoma Agraria Antonio Narro Buenavista, Saltillo COAH, 25315, M´exico E-mail: [email protected] 2
Centro de Investigaci´ on en Matem´ aticas, Apartado Postal 402 Guanajuato, GTO, 36000, M´exico E-mail: [email protected] (Received February 19, 2007; Accepted November 27, 2007)
Abstract This note concerns the asymptotic behavior of a Markov process obtained from normalized products of independent and identically distributed random matrices. The weak convergence of this process is proved, as well as the law of large numbers and the central limit theorem.
1. Introduction Motivated by the study of ergodic properties of dynamical systems, the analysis of the asymptotic behavior of products of random matrices can be traced back, at least, to the early sixties. Fundamental results were obtained in Furstenberg and Kesten (1960), Furstenberg (1963) and Oseledec (1968). The first of these papers considered a process {Mn } taking values in the space of k × k real matrices endowed with an appropriate norm · and, assuming that the Mn s are independent and identically distributed (iid), the authors studied the grow rate of the products Mn · · · M1 given by lim Mn Mn−1 · · · M1 1/n . n→∞
Mathematics subject classification number : 60F05, 60B10; 60B05. Key words and phrases: random products, Birkhoff’s distance, delayed process, invariant measure, law of large numbers, divisibility of a distribution, uniform differentiability of a family of characteristic functions. This work was supported by the PSF Organization under Grant No. 2005-7–02, and by the Consejo Nacional de Ciencia y Tecnolog´ıa under Grants 25357 and 61423. 0031-5303/2008/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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´ ´ R. CAVAZOS-CADENA and D. HERNANDEZ-HERN ANDEZ
It was proved that such a limit exists, and the law of large numbers as well as the central limit theorem were established; later on, Fustenberg and Kifer (1983) studied the asymptotic behavior of the vector norms Mn Mn−1 · · · M1 x1/n ,
x ∈ Rk .
In the fundamental papers mentioned above, the main issue was to study the asymptotic growth of the products Mn · · · M1 , and the theory of Lyapunov exponents was developed from logarithmic transformation of the products. In recent years, applications of products of random matrices in statistical physics, chaotic dynamical systems, filtering and Schrodinger operators has motivated a deep study of this theory; see, for instance, Cristiani, Paladin and Valpiani (1993), Atar and Zeitouni (1997), Bougerol and Lacroix (1985) and Carmona and Lacroix (1990). On the other hand, the assertion in Furstenberg and Kifer (1983) that ‘there are simple questions that are unanswered’, can be completed requiring also simple answers to simple questions. This work deals with some of those problems proving th
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