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Consider a $Ntimes n$ random matrix $Z_n=(Z^n_{j_1 j_2})$ where the individual entries are a realization of a properly rescaled stationary gaussian random field. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices such as $Z_n Z_n ^*$ and $(Z_n +A_n)(Z_n +A_n)^*$ where $A_n$ is a deterministic matrix with appropriate assumptions in the case where $nto infty$ and $frac Nn to c in (0,infty)$. The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs from related works in the literature (Boutet de Monvel et al., Girko, etc.).
Consider a $Ntimes n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $Y_{ij}^{n}=frac{sigma(i/N,j/n)}{sqrt{n}} X_{ij}^{n}$, the $X_{ij}^{n}$ being centered i.i.d. and $sigma:[0,1]^2 to (0,infty)$ being a continuous function called a
We prove the asymptotic independence of the empirical process $alpha_n = sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $beta_n = n (F_n(tau+frac{cdot}{n})-F_n(tau))$, where $F$ is an arbitrary cdf, differentiable at some point $
We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converg
This paper studies the local structure of continuous random fields on $mathbb R^d$ taking values in a complete separable linear metric space ${mathbb V}$. Extending seminal work of Falconer, we show that the generalized $(1+k)$-th order increment tan
We study the rate of convergence of the Mallows distance between the empirical distribution of a sample and the underlying population. The surprising feature of our results is that the convergence rate is slower in the discrete case than in the absol