Fixed points of multivariate smoothing transformations

在各种应用概率模型中，感兴趣量的极限分布满足递归分布方程，称为平滑变换的不动点方程，最突出的例子是中心正态分布所满足的卷积方程。虽然光滑变换及其不动点在一维空间中已经得到了广泛的研究，但在更高的维度上只有部分结果可用。该项目的目标是求解高维分布不动点方程的大类，了解不动点的重要特征，如密度的存在性、光滑性和衰减性或生存函数的尾部行为，并将所获得的理论结果应用于经典和最近的例子。

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.