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LINEAR STATISTICS OF POINT PROCESSES VIA ORTHOGONAL POLYNOMIALS
, 805
"... Abstract. For arbitrary β> 0, we use the orthogonal polynomials techniques developed in [10, 11] to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, ..."
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Abstract. For arbitrary β> 0, we use the orthogonal polynomials techniques developed in [10, 11] to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new. 1. Introduction and
LIMIT THEOREMS FOR ORTHOGONAL POLYNOMIALS RELATED TO CIRCULAR ENSEMBLES
, 2013
"... Abstract. For a natural extension of the circular unitary ensemble of order n, we study as n → ∞ the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescal ..."
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Abstract. For a natural extension of the circular unitary ensemble of order n, we study as n → ∞ the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescaling, we obtain a process indexed by t ∈ [0,1]. We show that it converges to a deterministic limit, and we describe the fluctuations and the large deviations. 1.