On a Generalisation of the Marcenko-Pastur Problem

We study the spectrum of generalized Wishart matrices, defined as \(\mathbf{F}=( X Y^\top + Y X^\top)/2T\), where \(X\) and \(Y\) are \(N \times T\) matrices with zero mean, unit variance IID entries and such that \(\mathbb{E}[X_{it} Y_{jt}]=c \delta_{i,j}\). The limit \(c=1\) corresponds to the Mar...

Celý popis

Uloženo v:

Podrobná bibliografie
Vydáno v:	arXiv.org
Hlavní autoři:	Bouchaud, Jean-Philippe, Potters, Marc
Médium:	Paper
Jazyk:	angličtina
Vydáno:	Ithaca Cornell University Library, arXiv.org 20.09.2020
Témata:	Cubic equations Eigenvalues
ISSN:	2331-8422
On-line přístup:	Získat plný text
Tagy:	Přidat tag Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!

Popis
Shrnutí:	We study the spectrum of generalized Wishart matrices, defined as \(\mathbf{F}=( X Y^\top + Y X^\top)/2T\), where \(X\) and \(Y\) are \(N \times T\) matrices with zero mean, unit variance IID entries and such that \(\mathbb{E}[X_{it} Y_{jt}]=c \delta_{i,j}\). The limit \(c=1\) corresponds to the Marcenko-Pastur problem. For a general \(c\), we show that the Stietjes transform of \(\mathbf{F}\) is the solution of a cubic equation. In the limit \(c=0\), \(T \gg N\) the density of eigenvalues converges to the Wigner semi-circle.
Bibliografie:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2009.07113