Lorden's inequality
Mathematics concept
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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.[1] Overshoots play a central role in renewal theory.[2]
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[2]
Proof
Three proofs are known due to Lorden,[1] Carlsson and Nerman[3] and Chang.[4]
