Lorden's inequality
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Short description: Mathematics concept
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]
- [math]\displaystyle{ \operatorname E (R_b) \leq \frac{\operatorname E (X^2)}{\operatorname E(X)}. }[/math]
Proof
Three proofs are known due to Lorden,[1] Carlsson and Nerman[3] and Chang.[4]
See also
References
- ↑ 1.0 1.1 Lorden, G. (1970). "On Excess over the Boundary". The Annals of Mathematical Statistics 41 (2): 520–527. doi:10.1214/aoms/1177697092.
- ↑ 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. 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. doi:10.2307/1427260.
- ↑ 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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