Lorden's inequality

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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]

Contents

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]

See also

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References

  1. 1 2 Lorden, G. (1970). "On Excess over the Boundary". The Annals of Mathematical Statistics. 41 (2): 520–527. doi: 10.1214/aoms/1177697092 . JSTOR   2239350.
  2. 1 2 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.
  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. doi: 10.2307/1427260 . JSTOR   1427260. S2CID   124416862.
  4. Chang, J. T. (1994). "Inequalities for the Overshoot". The Annals of Applied Probability. 4 (4): 1223. doi: 10.1214/aoap/1177004913 .