A DIFFERENCE PROPHET INEQUALITY FOR BOUNDED IID VARIABLES, WITH COST FOR OBSERVATIONS

• 成果类型: Article
• 署名作者: SAMUELCAHN, E
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989689
• 发表日期: 1992
• 页码: 1222-1228
• 关键词: stop rule

摘要

Let X(i) be i.i.d. random variables, 0 less-than-or-equal-to X(i) less-than-or-equal-to 1 and c greater-than-or-equal-to 0, and let Y(i) = X(i) - ic. It is shown that for all n, all c and all such X(i), E(max(i greater-than-or-equal-to 1 Y(i)) - sup(t) EY(t) < e-1, where t is a stopping rule and e-1 is shown to be the best bound for which the inequality holds. Specific bounds are also obtained for fixed n or fixed c. These results are very similar to those obtained by Jones for a similar problem, where 0 less-than-or-equal-to X(i) less-than-or-equal-to 1 are independent but not necessarily identically distributed. All results are valid and unchanged also when Y(i) is replaced by Y(i)* = max1 less-than-or-equal-to j less-than-or-equal-to i X(j) - ic.