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Let ${U^N_t}_{tge 0}$ be a standard Brownian motion on $mathbb{U}(N)$. For fixed $Ninmathbb{N}$ and $t>0$, we give explicit bounds on the $L_1$-Wasserstein distance of the empirical spectral measure of $U^N_t$ to both the ensemble-averaged spectral measure and to the large-$N$ limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.
The Brownian motion $(U^N_t)_{tge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{tge 0}$ as $Ntoinfty$. In this paper, we prove that it converges strongly as a process: not only in distribution but also
The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $mathsf{GL}(N;mathbb{C}),$ in the sense of $ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalu
This paper provides yet another look at the mixed fractional Brownian motion (fBm), this time, from the spectral perspective. We derive an approximation for the eigenvalues of its covariance operator, asymptotically accurate up to the second order. T
We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Itos type (Itos integral, Itos formula, Itos equation) through the corresponding G-Brownian motion. We will also present analytical c
We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $infty-$Wasserstein distance betwe