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 S_n = X₁ + X₂ + ... + X_n. Consider the first time S_n exceeds a given value b and at that time compute R_b = S_n − b. R_b is called the overshoot or excess at b. Lorden's inequality states that the expectation of this overshoot is bounded as

$\mathrm{E}(R_b)\le \frac{\mathrm{E}(X^2)}{\mathrm{E}(X)}.$

Proof

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

See also

• Wald's equation

References

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