We consider a product of 2 × 2 random matrices which appears in the physics literature in the analysis of some 1D disordered models. These matrices depend on a parameter ǫ > 0 and on a positive random variable Z. Derrida and Hilhorst (J Phys A 16:2641, 1983, §3) conjecture that the corresponding characteristic exponent has a regular expansion with respect to ǫ 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.2 Existence and first properties of the invariant measure X ǫ * We choose {0, 1} T N instead of {−1, 1} T N to simplify the formulas. They are equivalent by easy manipulations.

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