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.[1] Overshoots play a central role in renewal theory.[2]

Statement of inequality

Let X1, X2, ... be independent and identially 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 inquality states that the expectation of this overshoot is bounded as[2]

\operatorname E (R_b) \leq \frac{\operatorname E (X^2)}{\operatorname E(X)}.

Proof

Three proofs are known due to Lorden,[1] Carlsson and Nerman[3] and Chang.[4]

See also

References

  1. ↑ 1.0 1.1 Lorden, G. (1970). "On Excess over the Boundary". The Annals of Mathematical Statistics 41 (2): 520. doi:10.1214/aoms/1177697092. JSTOR 2239350.
  2. ↑ 2.0 2.1 Spouge, John L. (2007). "Inequalities on the overshoot beyond a boundary for independent summands with differing distributions". Statistics & Probability Letters 77 (14): 1486–1489. doi:10.1016/j.spl.2007.02.013.
  3. ↑ Carlsson, Hasse; Nerman, Olle (1986). "An Alternative Proof of Lorden's Renewal Inequality". Advances in Applied Probability (Applied Probability Trust) 18 (4): 1015–1016. JSTOR 1427260. Retrieved 19 Dec 2012.
  4. ↑ Chang, J. T. (1994). "Inequalities for the Overshoot". The Annals of Applied Probability 4 (4): 1223. doi:10.1214/aoap/1177004913.