Rate of convergence of linear functions on the unitary group

JP Keating, F Mezzadri, B Singphu

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)

Abstract

We study the rate of convergence to a normal random variable of the real and imaginary parts of , where U is an N × N random unitary matrix and AN is a deterministic complex matrix. We show that the rate of convergence is O(N−2 + b), with 0 ≤ b <1, depending only on the asymptotic behaviour of the singular values of AN; for example, if the singular values are non-degenerate, different from zero and O(1) as N → ∞, then b = 0. The proof uses a Berry–Esséen inequality for linear combinations of eigenvalues of random unitary matrices, and so appropriate for strongly dependent random variables.
Translated title of the contributionRate of convergence of linear functions on the unitary group
Original languageEnglish
Article number035204
Number of pages27
JournalJournal of Physics A: Mathematical and Theoretical
Volume44
Issue number3
DOIs
Publication statusPublished - Jan 2011

Bibliographical note

Publisher: IOP

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