The middle-scale asymptotics of Wishart matrices. (English) Zbl 1436.60019

Chételat, Didier; Wells, Martin T.

Ann. Stat. 47, No. 5, 2639-2670 (2019).

Summary: We study the behavior of a real \(p\)-dimensional Wishart random matrix with \(n\) degrees of freedom when \(n, p \rightarrow \infty\) but \(p/n \rightarrow 0\). We establish the existence of phase transitions when \(p\) grows at the order \(n^{(K+1)/(K+3)}\) for every \(K \in \mathbb{N}\), and derive expressions for approximating densities between every two phase transitions. To do this, we make use of a novel tool we call the \(\mathcal{F}\)-conjugate of an absolutely continuous distribution, which is obtained from the Fourier transform of the square root of its density. In the case of the normalized Wishart distribution, this represents an extension of the \(t\)-distribution to the space of real symmetric matrices.

MSC:

60B20	Random matrices (probabilistic aspects)
15B52	Random matrices (algebraic aspects)
60B10	Convergence of probability measures
60E10	Characteristic functions; other transforms

Keywords:

covariance estimation; Gaussian orthogonal ensemble; high-dimensional asymptotics; phase transitions; random matric theory

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