Lorden's inequality

In probability theory, Lorden's inequality is a bound for the moments of overshoot for a stopped sum of random variables, first published by Gary Lorden in 1970. Overshoots play a central role in renewal theory.

Statement of inequality
Let X1, X2, ... be independent and identically distributed positive random variables and define the sum Sn = X1 + X2 + ... + Xn. Consider the first time Sn exceeds a given value b and at that time compute Rb = Sn − b. Rb is called the overshoot or excess at b. Lorden's inequality states that the expectation of this overshoot is bounded as
 * $$\operatorname E (R_b) \leq \frac{\operatorname E (X^2)}{\operatorname E(X)}.$$

Proof
Three proofs are known due to Lorden, Carlsson and Nerman and Chang.