Fixed points of inhomogeneous smoothing transforms

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation $X \stackrel{d}{=} C + \sum_{i \geq 1} T_i X_i$,where$\stackrel{d}{=}$meansequalityindistribution,$(C,T_1,T_2,…)$isagivensequenceofnon-negativerandomvariablesand$X_1,X_2,…$ is a sequence of i.i.d.\ copies of the non-negative random variable $X$ independent of $(C,T_1,T_2,\mathrm{.}\mathrm{.}\mathrm{.})$. In this situation, $X$ (or, more precisely, the distribution of $X$) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Further, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C=0. Using this correspondence, we present a full characterization of the set of fixed points under mild assumptions.