29 research outputs found
Gaussian fluctuation for Gaussian Wishart matrices of overall correlation
In this note, we study the Gaussian fluctuations for the Wishart matrices
, where is
a random matrix whose entries are jointly Gaussian and correlated
with row and column covariance functions given by and respectively such
that . Under the assumptions and
, we establish the
convergence rate for the Wasserstein distance between a normalization of
and the corresponding Gaussian
ensemble. This rate is the same as the optimal one computed in
\cite{JL15,BG16,BDER16} for the total variation distance, in the particular
case where the Gaussian entries of are independent.
Similarly, we obtain the convergence rate for the
Wasserstein distance in the setting of random -tensors of overall
correlation. Our analysis is based on the Malliavin-Stein approach
Limiting behavior of large correlated Wishart matrices with chaotic entries
We study the fluctuations, as , of the Wishart matrix
associated to a random matrix with non-Gaussian
entries. We analyze the limiting behavior in distribution of
in two situations: when the entries of
are independent elements of a Wiener chaos of arbitrary order and when the
entries are partially correlated and belong to the second Wiener chaos. In the
first case, we show that the (suitably normalized) Wishart matrix converges in
distribution to a Gaussian matrix while in the correlated case, we obtain its
convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive
the rate of convergence in the Wasserstein distance via Malliavin calculus and
analysis on Wiener space
High dimensional regimes of non-stationary Gaussian correlated Wishart matrices
We study the high-dimensional asymptotic regimes of correlated Wishart
matrices , where is a
Gaussian random matrix with correlated and non-stationary entries. We prove
that under different normalizations, two distinct regimes emerge as both
and grow to infinity. The first regime is the one of central convergence,
where the law of the properly renormalized Wishart matrices becomes close in
Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the
second regime, a non-central convergence happens, and the law of the normalized
Wishart matrices becomes close in Wasserstein distance to that of the so-called
Rosenblatt-Wishart matrix recently introduced by Nourdin and Zheng. We then
proceed to show that the convergences stated above also hold in a functional
setting, namely as weak convergence in . As an
application of our main result (in the central convergence regime), we show
that it can be used to prove convergence in expectation of the empirical
spectral distributions of the Wishart matrices to the semicircular law. Our
findings complement and extend a rich collection of results on the study of the
fluctuations of Gaussian Wishart matrices, and we provide explicit examples
based on Gaussian entries given by normalized increments of a bi-fractional or
a sub-fractional Brownian motion