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 inequality states that the expectation of this overshoot is bounded as[2]
\( {\displaystyle \operatorname {E} (R_{b})\leq {\frac {\operatorname {E} (X^{2})}{\operatorname {E} (X)}}.} \)
Proof
Three proofs are known due to Lorden,[1] Carlsson and Nerman[3] and Chang.[4]
See also
Wald's equation
References
Lorden, G. (1970). "On Excess over the Boundary". The Annals of Mathematical Statistics. 41 (2): 520. doi:10.1214/aoms/1177697092. JSTOR 2239350.
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. PMC 2683021. PMID 19461943.
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.
Chang, J. T. (1994). "Inequalities for the Overshoot". The Annals of Applied Probability. 4 (4): 1223. doi:10.1214/aoap/1177004913.
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