Misplaced Pages

Lorden's inequality

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
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. 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 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

E ⁡ ( R b ) ≤ E ⁡ ( X 2 ) E ⁡ ( X ) . {\displaystyle \operatorname {E} (R_{b})\leq {\frac {\operatorname {E} (X^{2})}{\operatorname {E} (X)}}.}

Proof

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

See also

References

  1. ^ 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. ^ 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. 18 (4). Applied Probability Trust: 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.


Stub icon

This probability-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: