14 research outputs found
An Introduction to Wishart Matrix Moments
These lecture notes provide a comprehensive, self-contained introduction to
the analysis of Wishart matrix moments. This study may act as an introduction
to some particular aspects of random matrix theory, or as a self-contained
exposition of Wishart matrix moments. Random matrix theory plays a central role
in statistical physics, computational mathematics and engineering sciences,
including data assimilation, signal processing, combinatorial optimization,
compressed sensing, econometrics and mathematical finance, among numerous
others. The mathematical foundations of the theory of random matrices lies at
the intersection of combinatorics, non-commutative algebra, geometry,
multivariate functional and spectral analysis, and of course statistics and
probability theory. As a result, most of the classical topics in random matrix
theory are technical, and mathematically difficult to penetrate for non-experts
and regular users and practitioners. The technical aim of these notes is to
review and extend some important results in random matrix theory in the
specific context of real random Wishart matrices. This special class of
Gaussian-type sample covariance matrix plays an important role in multivariate
analysis and in statistical theory. We derive non-asymptotic formulae for the
full matrix moments of real valued Wishart random matrices. As a corollary, we
derive and extend a number of spectral and trace-type results for the case of
non-isotropic Wishart random matrices. We also derive the full matrix moment
analogues of some classic spectral and trace-type moment results. For example,
we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and
full matrix cases. Laplace matrix transforms and matrix moment estimates are
also studied, along with new spectral and trace concentration-type
inequalities
An introduction to Wishart matrix moments
© 2018 Now Publishers Inc. All rights reserved. These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics, non-commutative algebra, geometry, multivariate functional and spectral analysis, and of course statistics and probability theory. As a result, most of the classical topics in random matrix theory are technical, and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specific context of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory.We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary, we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example, we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied, along with new spectral and trace concentration-type inequalities
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
On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering
The purpose of this review is to present a comprehensive overview of the
theory of ensemble Kalman-Bucy filtering for linear-Gaussian signal models. We
present a system of equations that describe the flow of individual particles
and the flow of the sample covariance and the sample mean in continuous-time
ensemble filtering. We consider these equations and their characteristics in a
number of popular ensemble Kalman filtering variants. Given these equations, we
study their asymptotic convergence to the optimal Bayesian filter. We also
study in detail some non-asymptotic time-uniform fluctuation, stability, and
contraction results on the sample covariance and sample mean (or sample error
track). We focus on testable signal/observation model conditions, and we
accommodate fully unstable (latent) signal models. We discuss the relevance and
importance of these results in characterising the filter's behaviour, e.g. it's
signal tracking performance, and we contrast these results with those in
classical studies of stability in Kalman-Bucy filtering. We provide intuition
for how these results extend to nonlinear signal models and comment on their
consequence on some typical filter behaviours seen in practice, e.g.
catastrophic divergence
A perturbation analysis of stochastic matrix Riccati diffusions
Matrix differential Riccati equations are central in filtering and optimal
control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles
tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces