i.i.d.的预言不等式。具有随机到达时间的随机变量
Prophet inequalities for i.i.d. random variables with random arrival times
摘要：
假设$X_1,X_2,...$是i.i.d.。有限期望的非负随机变量，对于每个$k$，在具有单位速率的泊松过程的第$k$个到达时刻$S_k$处观察到$X_k$，该泊松过程与序列$\{X_k\}$无关。对于$t>0$，在期望的最大值$M(t):=\rE[\max_{k\geq 1} X_k \sI(S_k\leq t)]$和最优停止值$V(t):=\sup_{τ\in\TT}\sE[X_τ\sI(S_τ\leq t)]$之间进行比较，其中$\TT$是所有$\NN$值随机变量$τ$的集合，使得$\{τ=i\}$相对于由$(X_1,S_1),...,(X_i,S_i)$生成的$σ$代数是可测量的.例如，它表明，$M(t)/V(t)\leq 1+α_0$，其中$α_0\doteq 0.34149$满足$\int_0^1(y-y\ln y+α_0)^{-1} dy=1$，这个界限是渐近尖锐的$t\to\infty$。另一个结果是$M(t)/V(t)<2-(1-e^{-t})/t$，这个界限是渐近的，正如$t\downarrow 0$。上界的差异$M(t)-V(t)$也给出了，在额外的假设下，$X_k$有界。
英文摘要：
Suppose $X_1,X_2,...$ are i.i.d. nonnegative random variables with finite expectation, and for each $k$, $X_k$ is observed at the $k$-th arrival time $S_k$ of a Poisson process with unit rate which is independent of the sequence $\{X_k\}$. For $t>0$, comparisons are made between the expected maximum $M(t):=\rE[\max_{k\geq 1} X_k \sI(S_k\leq t)]$ and the optimal stopping value $V(t):=\sup_{τ\in\TT}\sE[X_τ\sI(S_τ\leq t)]$, where $\TT$ is the set of all $\NN$-valued random variables $τ$ such that $\{τ=i\}$ is measurable with respect to the $σ$-algebra generated by $(X_1,S_1),...,(X_i,S_i)$. For instance, it is shown that $M(t)/V(t)\leq 1+α_0$, where $α_0\doteq 0.34149$ satisfies $\int_0^1(y-y\ln y+α_0)^{-1} dy=1$; and this bound is asymptotically sharp as $t\to\infty$. Another result is that $M(t)/V(t)<2-(1-e^{-t})/t$, and this bound is asymptotically sharp as $t\downarrow 0$. Upper bounds for the difference $M(t)-V(t)$ are also given, under the additional assumption that the $X_k$ are bounded.
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