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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 variance profile. Consider now a deterministic $Ntimes n$ matrix $Lambda_n=(Lambda_{ij}^{n})$ whose non diagonal elements are zero. Denote by $Sigma_n$ the non-centered matrix $Y_n + Lambda_n$. Then under the assumption that $lim_{nto infty} frac Nn =c>0$ and $$ frac{1}{N} sum_{i=1}^{N} delta_{(frac{i}{N}, (Lambda_{ii}^n)^2)} xrightarrow[nto infty]{} H(dx,dlambda), $$ where $H$ is a probability measure, it is proven that the empirical distribution of the eigenvalues of $ Sigma_n Sigma_n^T$ converges almost surely in distribution to a non random probability measure. This measure is characterized in terms of its Stieltjes transform, which is obtained with the help of an auxiliary system of equations. This kind of results is of interest in the field of wireless communication.
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 eig
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 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
We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than
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