  We study spectrum of finite truncations of unbounded Jacobi matrices with
periodically modulated entries. In particular, we show that under some
hypotheses a sequence of properly normalized eigenvalue counting measures
converge vaguely to an explicit infinite Radon measure. To do so we link the
asymptotic behavior of the Christoffel-Darboux kernel on the diagonal with the
limiting measure. Finally, we derive strong asymptotics of the associated
orthogonal polynomials in the complex plane, which allows us to prove that
Cauchy transforms of the normalized eigenvalue counting measures converge
pointwise and which leads to a stronger notion of convergence.
