A Simple Expression for Mill's Ratio of the Student's t-Distribution

I show a simple expression of the Mill's ratio of the Student's t-Distribution. I use it to prove Conjecture 1 in Peter Auer,Nicoì o Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. We first need the following technical lemma. Hence we have x e x 2 2 √ e x 2 − 1 − 1 ≤ x 1 √ e x 2 − 1 ≤ 1, (2) where in the last inequality we used exp(z) − 1 ≥ z. The following theorem provides simple upper bounds to the Mill's ratio of a t-Student. Theorem 1. Let f ν (x) the pdf of a Student's t-distribution with ν degrees of freedom. Then, for any ν ≥ 0, we have +∞ a f ν (x)dx f ν (a) ≤ 1 + a 2 ν 1 2 + 1 √ ν , if a ≥ 0