arXiv subjects

Steen Thorbjørnsen

Publications and source records attributed to Steen Thorbjørnsen.

4 recordsLinked to original sources

arXiv2020

Completely Random Measures and Lévy Bases in Free probability

This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved that such random measures can be decomposed into the sum of a purely atomic part and a (freely) infinitely divisible part. The latter part (termed a free Lévy basis) is studied in detail in terms of the free Lévy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosinski is developed. Finally a Lévy-Itô type decomposition for general free Lévy bases is established.

math.PR↗

arXiv2017

Uffe Haagerup - His life and mathematics

In remembrance of Professor Uffe Valentin Haagerup (1949--2015), as a brilliant mathematician, we review some aspects of his life, and his outstanding mathematical accomplishments.

math.FA↗

arXiv2017

The normal distribution is freely selfdecomposable

The class of selfdecomposable distributions in free probability theory was introduced by Barndorff-Nielsen and the third named author. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this paper, we prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey-Wimp-Kerov distribution $μ_c$ is freely selfdecomposable for any $c$ in $[-1,0]$. The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform.

math.PR↗

arXiv2010

Asymptotic expansions for the Gaussian Unitary Ensemble

Let g:{\mathbb R} --> {\mathbb C} be a C^{\infty}-function with all derivatives bounded and let tr_n denote the normalized trace on the n x n matrices. In the paper [EM] Ercolani and McLaughlin established asymptotic expansions of the mean value E{tr_n(g(X_n))} for a rather general class of random matrices X_n,including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a GUE random matrix X_n that E{tr_n(g(X_n))}= \frac{1}{2π}\int_{-2}^2 g(x)\sqrt{4-x^2} dx +\sum_{j=1}^k\frac{α_j(g)}{n^{2j}}+ O(n^{-2k-2}), where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients α_j(g), j\in{\mathbb N}, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Tr_n[f(X_n)],Tr_n[g(X_n)]}, where f is a function of the same kind as g, and Tr_n=n tr_n. Special focus is drawn to the case where g(x)=1/(z-x) and f(x)=1/(w-x) for non-real complex numbers z and w. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the GUE(n,1/n).

math.PR↗