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Register a new userConvergence of the empirical spectral measure of unitary Brownian motion
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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.
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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 in operator norm. In particular, for a fixed time $t>0$, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.
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 eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $bar{Sigma}_{t},$ which is strictly positive and real analytic on $Sigma_{t}$. This density has a simple form in polar coordinates: [ W_{t}(r,theta)=frac{1}{r^{2}}w_{t}(theta), ] where $w_{t}$ is an analytic function determined by the geometry of the region $Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a shadow of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigners semicircle law and Ginibres circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.
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. This in turn allows to compute the exact $L_2$-small ball probabilities, previously known only at logarithmic precision. The obtained expressions show an interesting stratification of scales, which occurs at certain values of the Hurst parameter of the fractional component. Some of them have been previously encountered in other problems involving such mixtures.
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 calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty. Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y. Two important theorems have been proved: The law of large number and the central limit theorem.
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 between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slepv{c}ev to the case that the true distribution has an unbounded density.
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