Universal Bounds and Monotonicity Properties of Ratios of Hermite and Parabolic Cylinder Functions
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. Diff erently 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 infi nitesimal generator of the Ornstein-Uhlenbeck process. As a byproduct, we obtain TurĂ¡n type inequalities.
615
Center for Mathematical Economics
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