Convexity of the Lyapunov exponent
Let \(\gamma(p)\) be the maximal Lyapunov exponent for an i.i.d. random product of two matrices \(A\) and \(B\) where the factor \(A\) occurs with probability \(p\) and \(B\) occurs with probability 1-p. Under suitable conditions on \(A\) and \(B\) convexity or concavity of \(\gamma\) on \((0,1)\) is proved while general results cannot hold. F.ex. it is not true that \(\gamma\) is convex on \((0,1)\) for \(A\) and \(B\) normal. Typical positive results are: If \(A\) and \(B\) are symmetric positive-semidefinite \(2\times 2\) matrices then \(\gamma\) is convex. If \(A\) is a nonnegative \(2\times 2\) matrix with nonnegative determinant and \(B\) is the transpose of \(A\) then \(\gamma\) is concave on \((0,1)\).
scientific article; zbMATH DE number 1371390
scientific article