Large deviations for products of random matrices and applications in time series analysis

SPEAKER:  Sebastian Mentemeier, University of Wroclaw

ABSTRACT:  The recursive equation, for example, of a squared ARCH(p) process can be written in the form X(n) = A(n) X(n-1) + B(n), where X(n) is a p-dimensional vector (the first entry being the state of the process at time n), while A(n) is an i.i.d. random matrix and B(n) is an i.i.d. random vector, each with nonnegative entries, which are given by the coefficients of the ARCH(p) process.
  Following classical works, I will show how the existence of a (strong) stationary distribution for the ARCH(p) process can be deduced from a strong law of large numbers for products of random matrices, namely the Furstenberg-Kesten theorem.
  Subsequently, I will introduce a corresponding large deviation result for products of random matrices, which allows one to prove heavy tail properties of the stationary distribution without applying Kesten's renewal theorem.  (Joint work with Dariusz Buraczewski, Univ. Wroclaw.)