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 [mu]: W[congruent with][summation operator]j=1[infinity] TjWj, where W,W1,W2,... have distribution [mu] and T,W1,W2,... are independent. The corresponding functional equation is [phi](t)=E [product operator]j=1[infinity] [phi](tTj), where [phi] 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 -Lebesgue densities. The article extends results on the case of non-negative T and non-negative solutions.