Matrix refinement type equations

Taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the Fourier transform of an L1-solution of the following random matrix refinement type equationf(x)=∫Ω|detL(ω)|C(ω)f(L(ω)x-M(ω))P(dω),where M:Ω→Rp is a random vector and L:Ω→Rp×p,C:Ω→Rq×q are random matrices. Moreover, the dimension of the space of all L1-solutions of this refinement type equation is examined.