Fixed points of multivariate smoothing transformations

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In various models of applied probability, the limiting distributions of quantities of interest satisfy recursive distributional equations called fixed-point equations of smoothing transformations, the most prominent example being the convolution equation satisfied by the centered normal distributions. While smoothing transformations and their fixed points have been studied extensively in one dimension, only partial results are available in higher dimensions. The goal of the project is to solve large classes of distributional fixed-point equations in higher dimensions, to understand important features of fixed points such as existence, smoothness and decay of densities or the tail behavior of the survival function, and to apply the theoretical results obtained to classical and recent examples.