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Register a new userThe empirical eigenvalue distribution of a Gram matrix: From independence to stationarity
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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.).
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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.
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 $tau$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n in N$, there is a deterministic correspondence between $alpha_n$ and $beta_n$. Precisely, we show that the pair $(alpha_n,beta_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge and a two-sided Poisson process. Since these processes have jumps, in particular if $F$ itself has jumps, the Skorokhod product space $D(R) times D(R)$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D(R) times D(R)$ by establishing the classical principle, devised by Yu. V. Prokhorov, that finite-dimensional convergence and tightness imply weak convergence. Several tightness criteria are given. Finally, the convergence of the pair $(alpha_n,beta_n)$ implies convergence of each of its components, thus, in passing, we provide a thorough proof of these known convergence results in a very general setting. In fact, the condition on $F$ to be differentiable in at least one point is only required for $beta_n$ to converge and can be further weakened.
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 $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M times N$ random matrix whose entries are independent real or complex random variables, assuming that both $M$ and $N$ tend to infinity at a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comparison method [27] using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.
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 tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of ${mathbb V}$-valued intrinsic random functions of order $k$ (IRF$_k$, $k=0,1,cdots$). To this end, we focus on the special case where ${mathbb V}$ is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order ${mathbb V}$-valued IRF$_k$s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar ${mathbb V}$-valued IRF$_k$s, generalizing results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These processes are the Hilbert-space-valu
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 absolutely continuous setting. We show how the hazard function plays a significant role in these calculations. As an application, we recall that the quantity studied provides an upper bound on the distance between the bootstrap distribution of a sample mean and its true sampling distribution. Moreover, the convenient properties of the Mallows metric yield a straightforward lower bound, and therefore a relatively precise description of the asymptotic performance of the bootstrap in this problem.
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