Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line

We study ratio asymptotics, that is, existence of the limit of Pn+1(z)/P-n(z) (P-n = monic orthogonal polynomial) and the existence of weak limits of p(n)(2)dmu (p(n) = P-n/parallel toP(n)parallel to) as n --> infinity for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z(0) with Im(z(0)) not equal 0 implies dmu is in a Nevai class (i.e., a(n) --> a and b(n) --> b where a(n), b(n) are the off-diagonal and diagonal Jacobi parameters). For mu's with bounded support, we prove p(n)(2)dmu has a weak limit if and only if lim b(n), lim a(2n), and lim a(2n+1) all exist. In both cases, we write down the limits explicitly. (C) 2003 Elsevier Inc. All rights reserved.