Multi-dimensional smoothing transformations: Existence, regularity and stability of fixed points

Federico Bassetti and Daniel Matthes

Stochastic Processes and their Applications, 2014, vol. 124, issue 1, 154-198

Abstract: We analyze a class of smoothing transformations on probability measures in multiple space dimensions. Applying a synthesis of probabilistic methods and Fourier analysis, we prove existence and uniqueness of a fixed point inside the class of probability measures of finite second moment, characterize it as a scale mixture of Gaussians, and discuss its regularity. We also classify its tail, which might be of Pareto type. As an application, we study the stability of stationary solutions in a Kac-type kinetic model. In particular, we prove that the domain of attraction is precisely the probability measures of finite second moment.

Keywords: Multi-dimensional smoothing transformations; Central limit theorems; Mixture of Gaussians; Fourier-based metric; Kac model (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1016/j.spa.2013.07.006

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