  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.

Probability