Inequalities for the Overshoot

Let X1,X2,. be independent and identically distributed positive random variables with Sn=X1+.+Xn, and for nonnegative b define Rb=inf{Sn.b:Sn>b}. Then Rb is called the overshoot at b. In terms of the moments of X1, Lorden gave bounds for the moments of Rb that hold uniformly over all b. Using a coupling argument, we establish stochastic ordering inequalities that imply the moment inequalities of Lorden. In addition to simple new proofs of Lorden's inequalities, we provide new inequalities for the tail probabilities P{Rb>x} and moments of Rb that improve upon those of Lorden. We also present conjectures for sharp moment inequalities and describe an application to the first ladder height of random walks.