“Lorden's inequality”的意思、由来-开放百科全书

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 S_n = X₁ + X₂ + ... + X_n. Consider the first time S_n exceeds a given value b and at that time compute R_b = S_n − b. R_b 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

References

1. ^¹{{Cite journal | last1 = Lorden | first1 = G. | title = On Excess over the Boundary | doi = 10.1214/aoms/1177697092 | journal = The Annals of Mathematical Statistics | volume = 41 | issue = 2 | pages = 520 | year = 1970 | pmid = | pmc = | jstor = 2239350}}
2. ^¹{{Cite journal | last1 = Spouge | first1 = John L. | title = Inequalities on the overshoot beyond a boundary for independent summands with differing distributions | doi = 10.1016/j.spl.2007.02.013 | journal = Statistics & Probability Letters | volume = 77 | issue = 14 | pages = 1486–1489 | year = 2007 | pmid = | pmc = 2683021}}
3. ^{{cite journal | last1 = Carlsson | first1 = Hasse | last2 = Nerman | first2 = Olle | year = 1986 | title = An Alternative Proof of Lorden's Renewal Inequality | journal = Advances in Applied Probability | volume = 18 | issue = 4 | pages = 1015–1016 | publisher = Applied Probability Trust |jstor=1427260}}
4. ^{{Cite journal | last1 = Chang | first1 = J. T. | title = Inequalities for the Overshoot | doi = 10.1214/aoap/1177004913 | journal = The Annals of Applied Probability | volume = 4 | issue = 4 | pages = 1223 | year = 1994 | pmid = | pmc = }}