There are three real regions for the asymptotics of orthogonal polynomials: (1) on the spectrum of the measure, where the polynomials oscillate; (2) off the convex hull of the points of the spectrum, where the polynomials have the same sign for real values of the variable; and (3) on the convex hull of the spectrum but not on the spectrum, which may be a void set. The authors consider Jacobi, Laguerre, Meixner and Charlier polynomials where the parameters (except c for Meixner polynomials) are integers, and give a probabilistic proof for asymptotic formulas in the second case above. For very special values of the parameters they obtain asymptotic series on the same set.