Precise Large Deviation Results for Products of Random Matrices

Dariusz Buraczewski; Sebastian Mentemeier

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arXiv · 1405.6505

Precise Large Deviation Results for Products of Random Matrices

Dariusz Buraczewski·Sebastian Mentemeier

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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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Dariusz Buraczewski, Sebastian Mentemeier. 2014-05-26. Precise Large Deviation Results for Products of Random Matrices. https://arxiv.org/abs/1405.6505

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