High-dimensional limits of eigenvalue distributions for general Wishart process

In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.Comment: 28 page

On spectrum of sample covariance matrices from large tensor vectors

In this paper, we study the limiting spectral distribution of sums of independent rank-one large $k$-fold tensor products of large $n$-dimensional vectors. In the literature, the limiting moment sequence is obtained for the case $k=o(n)$ and $k=O(n)$. Under appropriate moment conditions on base vectors, it has been showed that the eigenvalue empirical distribution converges to the celebrated Mar\v{c}enko-Pastur law if $k=O(n)$ and the components of base vectors have unit modulus, or $k=o(n)$. In this paper, we study the limiting spectral distribution by allowing $k$ to grow much faster, whenever the components of base vectors are complex random variables on the unit circle. It turns out that the limiting spectral distribution is Mar\v{c}enko-Pastur law. Comparing with the existing results, our limiting setting only requires $k \to \infty$. Our approach is based on the moment method.Comment: 20 pages, 7 figure

On a class of stochastic fractional heat equations

For the fractional heat equation $\frac{\partial}{\partial t} u(t,x) = -(-\Delta)^{\frac{\alpha}{2}}u(t,x)+ u(t,x)\dot W(t,x)$ where the covariance function of the Gaussian noise $\dot W$ is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that $d<2+\alpha$ is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field

On spectrum of sample covariance matrices from large tensor vectors

In this paper, we study the limiting spectral distribution of sums of independent rank-one large $k$-fold tensor products of large $n$-dimensional vectors. In the literature, the limiting moment sequence is obtained for the case $k=o(n)$ and $k=O(n)$. Under appropriate moment conditions on base vectors, it has been showed that the eigenvalue empirical distribution converges to the celebrated Mar\v{c}enko-Pastur law if $k=O(n)$ and the components of base vectors have unit modulus, or $k=o(n)$. In this paper, we study the limiting spectral distribution by allowing $k$ to grow much faster, whenever the components of base vectors are complex random variables on the unit circle. It turns out that the limiting spectral distribution is Mar\v{c}enko-Pastur law. Comparing with the existing results, our limiting setting only requires $k \to \infty$. Our approach is based on the moment method

Stochastic partial differential equations associated with Feller processes

For the stochastic partial differential equation $\frac{\partial u}{\partial t}=\mathcal L u +u\dot W$ where $\dot W$ is Gaussian noise colored in time and $\mathcal L$ is the infinitesimal generator of a Feller process $X$, we obtain Feynman-Kac type of representations for the Stratonovich and Skorohod solutions as well as for their moments. The regularity of the law and the H\"older continuity of the solutions are also studied.Comment: 30 page

On spectral distribution of sample covariance matrices from large dimensional and large k-fold tensor products

We study the eigenvalue distributions for sums of independent rank-one k-fold tensor products of large n-dimensional vectors. Previous results in the literature assume that k=o(n) and show that the eigenvalue distributions converge to the celebrated Marčenko-Pastur law under appropriate moment conditions on the base vectors. In this paper, motivated by quantum information theory, we study the regime where k grows faster, namely k=O(n). We show that the moment sequences of the eigenvalue distributions have a limit, which is different from the Marčenko-Pastur law, and the Marčenko-Pastur law limit holds if and only if k=o(n) for this tensor model. The approach is based on the method of moments

Hyperbolic Anderson model with time-independent rough noise: Gaussian fluctuations

In this article, we study the hyperbolic Anderson model in dimension 1, driven by a time-independent rough noise, i.e. the noise associated with the fractional Brownian motion of Hurst index $H \in (1/4,1/2)$. We prove that, with appropriate normalization and centering, the spatial integral of the solution converges in distribution to the standard normal distribution, and we estimate the speed of this convergence in the total variation distance. We also prove the corresponding functional limit result. Our method is based on a version of the second-order Gaussian Poincar\'e inequality developed recently in [27], and relies on delicate moment estimates for the increments of the first and second Malliavin derivatives of the solution. These estimates are obtained using a connection with the wave equation with delta initial velocity, a method which is different than the one used in [27] for the parabolic Anderson model