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## 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 contribution | Rate of convergence of linear functions on the unitary group |
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Original language | English |

Article number | 035204 |

Number of pages | 27 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 44 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 2011 |

### Bibliographical note

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## Projects

- 1 Finished