Center for Mathematical Economics Working Papers No. 615
We obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions. Those ratios are shown to be strictly decreasing and bounded by universal constants. Differently to usual analytic approaches, we employ simple purely probabilistic arguments to derive our results. In particular, we exploit the relation between Hermite and parabolic cylinder functions and the eigenfunctions of the infinitesimal generator of the Ornstein-Uhlenbeck process. As a byproduct, we obtain Turán type inequalities.
Hermite functions parabolic cylinder functions Turán type inequalities Ornstein-Uhlenbeck process