Let θ1,…,θn be random variables from Dyson's circular
β-ensemble with probability density function Const⋅∏1≤j<k≤n∣eiθj−eiθk∣β. For
each n≥2 and β>0, we obtain some inequalities on
E[pμ(Zn)pν(Zn)ˉ], where
Zn=(eiθ1,…,eiθn) and pμ is the power-sum
symmetric function for partition μ. When β=2, our inequalities
recover an identity by Diaconis and Evans for Haar-invariant unitary matrices.
Further, we have the following: limn→∞E[pμ(Zn)pν(Zn)ˉ]=δμν(β2)l(μ)zμ for any β>0 and
partitions μ,ν; limm→∞E[∣pm(Zn)∣2]=n for any
β>0 and n≥2, where l(μ) is the length of μ and zμ is
explicit on μ. These results apply to the three important ensembles: COE
(β=1), CUE (β=2) and CSE (β=4). We further examine the
nonasymptotic behavior of E[∣pm(Zn)∣2] for β=1,4. The
central limit theorems of ∑j=1ng(eiθj) are obtained when (i)
g(z) is a polynomial and β>0 is arbitrary, or (ii) g(z) has a Fourier
expansion and β=1,4. The main tool is the Jack function.Comment: Published at http://dx.doi.org/10.1214/14-AOP960 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org