Precise Large Deviation Results for Products of Random Matrices

2014/05/26 by Buraczewski, Dariusz, Mentemeier, Sebastian
#FOS: Mathematics #Primary 60F10 #Probability (math.PR) #secondary 60H25

paper · doi:10.48550/arxiv.1405.6505

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.

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