The Maclaurin Series for (Max,+) Lyapunov Exponents:
The Bernoulli Case

Bernd Heidergott
Delft University of Technology

ALAPEDES MEETING
March 31, 1999

We obtain the Maclaurin series for the Lyapunov exponent $\lambda( \theta )$ of a sequence of independently and identically distributed random matrices over the $( \max ,+ )$ semiring, generated via a Bernoulli $\theta$ scheme. The key assumption is that one of the possible outcomes of the random matrices has an eigenspace of dimension one. We apply the theory of weak differentiation to show that $\lambda( \theta )$ is analytical on [ 0 , 1). We show that the Maclaurin series converges to $\lambda( \theta )$on [ 0 , p ] for any p < 1. This result holds for the transient case as well. We will provide an intuitive explanation why we cannot expect the Maclaurin series to converge on the whole unit interval.