Fixed points of multivariate smoothing transforms with scalar weights

Given a sequence $(C_1,\ldots,C_d,T_1,T_2,\ldots)$ of real-valued random variables with $N := \#\{j \geq 1: T_j \not = 0\} < \infty$ almost surely, there is an associated smoothing transformation which maps a distribution $P$ on $\mathbb{R}^d$ to the distribution of $\sum_{j \geq 1} T_j \mathbf{X}^{(j)} + \mathbf{C}$ where $\mathbf{C} = (C_1,\ldots,C_d)$ and $(\mathbf{X}^{(j)})_{j \geq 1}$ is a sequence of independent random vectors with distribution $P$ independent of $(C_1,\ldots,C_d,T_1,T_2,\ldots)$. We are interested in the fixed points of this mapping. By improving on the techniques developed in [G. Alsmeyer, J.D. Biggins, and M. Meiners. The functional equation of the smoothing transform {\em Ann. Probab.}, 40(5):2069--2105, 2012] and [G. Alsmeyer and M. Meiners. Fixed points of the smoothing transform: two-sided solutions. {\em Probab. Theory Related Fields}, 155(1-2):165--199, 2013], we determine the set of all fixed points under weak assumptions on $(C_1,\ldots,C_d,T_1,T_2,\ldots)$. In contrast to earlier studies, this includes the most intricate case when the $T_j$ take both positive and negative values with positive probability. In this case, in some situations, the set of fixed points is a subset of the corresponding set when the $T_j$ are replaced by their absolute values, while in other situations, additional solutions arise.