Maharam extension and stationary stable processes
نویسنده
چکیده
We give a second look at stationary stable processes by interpreting the self-similar property at the level of the Lévy measure as characteristic of a Maharam system. This allows us to derive structural results and their ergodic consequences. 1. Introduction. In a fundamental paper [9], Rosi´nski revealed the hidden structure of stationary symmetric α-stable (SαS) processes. Namely, he proved that, following Hardin [5], through what is called a minimal spectral representation, such a process is driven by a nonsingular dynamical system. Such a result was proved to classify those processes according to their ergodic properties such as various kinds of mixing. In [13], we used a different approach as we considered the whole family of stationary infinitely divisible processes without Gaussian part (called IDp processes). The key tool there was the Lévy measure system of the process, which was measure-preserving and not just merely nonsingular. So far, in the stable case, the connection between the Lévy measure and the nonsingular system was not clear. This is the purpose of this paper, to fill the gap and go beyond both approaches. Indeed, we will prove that Lévy measure systems of α-stable processes have the form of a so-called Maharam system. This observation has some interesting consequences as it allows us to derive very quickly minimal spectral representations in the SαS case, to reinforce factorization results, and to refine ergodic classification. Let us explain very loosely the mathematical features of stable distributions we will be using. Observe that stable distributions are characterized by a self-similar property which is obvious when observing the corresponding Lévy process: If X t is an α-stable Lévy process, then b −1/α X bt has the same distribution.
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تاریخ انتشار 2012