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.
Let X<sub>1</sub>, X<sub>2</sub>, ... be independent and identically distributed positive random variables and define the sum S<sub>n</sub> = X<sub>1</sub> + X<sub>2</sub> + ... + X<sub>n</sub>. Consider the first time S<sub>n</sub> exceeds a given value b and at that time compute R<sub>b</sub> = S<sub>n</sub> â b. R<sub>b</sub> is called the overshoot or excess at b. Lorden's inequality states that the expectation of this overshoot is bounded as
Three proofs are known due to Lorden, Carlsson and Nerman and Chang.