Conference Agenda

We investigate linear spectral statistics (LSS) of a sample correlation matrix $R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that LSS satisfy a central limit theorem (CLT) and the centering and scaling sequences are universal in the sense that they do not depend on the entry distribution. Under a symmetry and a regular variation assumption with index $\alpha$ and any growth rate of the dimension, we prove that the universal CLT remains valid for $\alpha >3$. Moreover, for $\alpha\le 3$ we establish a non-universal CLT with norming sequences depending on the value of $\alpha$. Our findings are illustrated in a small simulation study.

We prove a central limit error bound for convolution powers of laws with finite moments of order $ r \in \mathopen]2,3\mathclose] $, taking a closeness of the laws to normality into account:

Going beyond the Berry (1941) - Esseen (1942) theorem, for Lyapunov's (1901) limit theorem the classical error bound with a constant independent of $r$ was first obtained explicitly by Katz (1963). Up to a universal constant, our result sharpens the i.i.d. case and generalises the special case of $r=3$ obtained by Mattner (2024).

The hermitian Brownian motion is one of the most studied random matrix processes and its eigenvalue process is the well known Dyson Brownian motion. It turns out that deforming this matrix model by adding a certain deterministic diagonal matrix, which is linearly scaled by time t, gives rise to another interesting eigenvalue process. We we will see that the latter is precisely the componentwise logarithm of the singular value process of the multiplicative Brownian motion on $\operatorname{GL}(N,\mathbb{C})$. In the large $N$ limit, we in turn get a formula relating the free additive convolution of the semicircle and uniform distribution to the distribution of the free positive multiplicative Brownian motion. Based on joint work with Michael Voit.

Based on a recently developed theory for strongly continuous convex monotone semigroups on spaces of continous functions, we provide a central limit theorem for convex expectations including explicit convergence rates. In the present context, the nonlinearity of the expectation arises due to uncertainty regarding the distribution of the samples which transfers to the transition probabilities determining the corresponding semigroup. Our results are consistent with Peng's G-framework for sublinear expectations and, to the best of our knowledge, the first extension to the convex case. This is of particular interest for applications in mathematical finance. Furthermore, the framework allows to consider large deviations results as a law of larges numbers for suitable convex expectations.