An Introduction to Wishart Matrix Moments

Bishop, Adrian N.; Moral, Pierre Del; Niclas, Angèle

doi:10.1561/9781680835076

Cited by 3 publications

(2 citation statements)

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“…The fluctuation analysis of this third class of EnKF model can also be developed easily by combining the Lipschitz-type estimates w.r.t. the initial state presented in [9,10], with conventional sample mean error estimates based on independent copies of the initial values, see for instance [13] for χ 2 -type estimates associated with sample covariance estimates.…”

Section: Mean Field Particle Interpretationsmentioning

confidence: 99%

On one-dimensional Riccati diffusions

Bishop¹,

Moral²,

Kamatani³

et al. 2019

Ann. Appl. Probab.

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This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration, and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.

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Section: Mean Field Particle Interpretationsmentioning

confidence: 99%

On one-dimensional Riccati diffusions

Bishop¹,

Moral²,

Kamatani³

et al. 2019

Ann. Appl. Probab.

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“…Random matrix theory plays a fundamental role in many areas of mathematics, either theoretical ones such as non-commutative algebra, combinatorics, geometry or spectral analysis, or applied ones such as statistical physics, signal processing or multivariate analysis and statistical theory. In the latter, one type of random matrices are particularly important as they are used to model, for example, sample covariance matrices (see e.g., the surveys [9,18,26]), which in the era of data driven analysis have now a growing importance in practice. This type of matrices are called Wishart matrices and have been introduced by the statistician John Wishart in [30].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

High dimensional regimes of non-stationary Gaussian correlated Wishart matrices

Bourguin,

Dang

2020

Preprint

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We study the high-dimensional asymptotic regimes of correlated Wishart matrices d −1 YY T , where Y is a n × d Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both n and d 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 C ([a, b]; Mn(R)). 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.

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