Precise Large Deviation Results for Products of Random Matrices

TL;DR

This paper extends large deviation theory to products of random matrices, providing precise estimates and Edgeworth expansions for the distribution of vector norms, with applications in financial time series analysis.

Contribution

It generalizes the Bahadur-Rao theorem to matrix products, offering third-order Edgeworth expansions and precise large deviation results using the Nagaev-Guivarch method.

Findings

Derived precise large deviation estimates for matrix product norms

Established third-order Edgeworth expansions for the distribution of vector norms

Applied results to matrix recursions in financial time series

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.