Convexity properties of products of random nonnegative matrices

Consider a sequence of N x N random nonnegative matrices in which each element depends on a vector u of parameters. The nth partial product is the random matrix formed by multiplying, from right to left, the first n of these random matrices in order. Under certain conditions, the elements of the nth partial product grow asymptotically exponentially as n increases, and the logarithms of the discrete long-run growth rates are convex functions of u. These conditions are met by some models in statistical mechanics and demography. Consequently, the Helmholtz free energy is concave and the population growth rate is convex in these models.