Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Precise large deviation results for products of random matrices

Dariusz Buraczewski and Sebastian Mentemeier

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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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