Fixed points with finite variance of a smoothing transformation

Let T=(T1,T2,T3,…) be a sequence of real random variables. We investigate the following fixed point equation for distributions μ: W≅∑j=1∞ TjWj, where W,W1,W2,… have distribution μ and T,W1,W2,… are independent. The corresponding functional equation is φ(t)=E ∏j=1∞ φ(tTj), where φ is a characteristic function. We consider solutions of the fixed point equation with finite variance. Results about existence and uniqueness are derived. In the situation of solutions with zero expectation we give a representation of the characteristic functions of solutions and treat the question of moments and C∞-Lebesgue densities. The article extends results on the case of non-negative T and non-negative solutions.