Regularity and the Cesaro-Nevai class

We consider OPRL and OPUC with measures regular in the sense of Ullman-Stahl-Totik and prove consequences on the Jacobi parameters or Verblunsky coefficients. For example, regularity on $[-2,2]$ implies ${\lim }_{N\to \mathrm{\infty }}{N}^{-1}[{\sum }_{n=1}^N(a_n-1{)}^2+b_n^2]=0$.