We construct explicit invariant measures for a family of infinite products of
random, independent, identically-distributed elements of SL(2,C). The matrices
in the product are such that one entry is gamma-distributed along a ray in the
complex plane. When the ray is the positive real axis, the products are those
associated with a continued fraction studied by Letac and Seshadri [Z. Wahr.
Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the
continued fraction is a generalised inverse Gaussian. We extend this result by
finding the distribution for an arbitrary ray in the complex right-half plane,
and thus compute the corresponding Lyapunov exponent explicitly. When the ray
lies on the imaginary axis, the matrices in the infinite product coincide with
the transfer matrices associated with a one-dimensional discrete Schroedinger
operator with a random, gamma-distributed potential. Hence, the explicit
knowledge of the Lyapunov exponent may be used to estimate the (exponential)
rate of localisation of the eigenstates.