We consider a product of 2 x 2 random matrices which appears in the physics literature
in the analysis of some 1D disordered models. These matrices depend on a parameter
? >0 and on a positive random variable Z. Derrida and Hilhorst (J. Phys. 16(12), 2641,
1983, 3) conjecture that the corresponding characteristic exponent has a regular expansion
with respect to ? up to and not further an order determined by the distribution of
Z. We give a rigorous proof of that statement. We also study the singular term which
breaks that expansion.