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Dette, Holger; Tomecki, Dominik; Venker, Martin

Universality in random moment problems. (English) Zbl 1390.60084

Electron. J. Probab. 23, Paper No. 15, 23 p. (2018).
Summary: Let \(\mathcal{M}_n(E)\) denote the set of vectors of the first \(n\) moments of probability measures on \(E\subset \mathbb{R}\) with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in the recent literature. For instance, it has been shown that a uniformly distributed moment sequence in \(\mathcal{M}_n([0,1])\) converges in the large \(n\) limit to the moment sequence of the arcsine distribution. In this article we provide a unifying viewpoint by identifying classes of more general distributions on \(\mathcal{M}_n(E)\) for \(E=[a,b]\), \(E=\mathbb{R}_+\) and \(E=\mathbb{R}\), respectively, and discuss universality problems within these classes. In particular, we demonstrate that the moment sequence of the arcsine distribution is not universal for \(E\) being a compact interval. Rather, there is a universal family of moment sequences of which the arcsine moment sequence is one particular member. On the other hand, on the moment spaces \(\mathcal{M}_n(\mathbb{R}_+)\) and \(\mathcal{M}_n(\mathbb{R})\) the random moment sequences governed by our distributions exhibit for \(n\rightarrow \infty\) a universal behaviour: The first \(k\) moments of such a random vector converge almost surely to the first \(k\) moments of the Marchenko-Pastur distribution (half line) and Wigner’s semi-circle distribution (real line). Moreover, the fluctuations around the limit sequences are Gaussian. We also obtain moderate and large deviations principles and discuss relations of our findings with free probability.

Cited in 2 Documents

MSC:

60F05 Central limit and other weak theorems
30E05 Moment problems and interpolation problems in the complex plane
60B20 Random matrices (probabilistic aspects)

Keywords:

random moment sequences; universality; CLT; large deviations principles; Stieltjes transform; free probability
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Full Text: DOI arXiv Euclid

References:

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