  We study Nevai's condition from the theory of orthogonal polynomials on the
real line. We prove that a large class of measures with unbounded Jacobi
parameters satisfies Nevai's condition locally uniformly on the support of the
measure away from a finite explicit set. This allows us to give applications to
relative uniform and weak asymptotics of Christoffel-Darboux kernels on the
diagonal and to limit theorems for unconventionally normalized global linear
statistics of orthogonal polynomial ensembles.
