Comparisons of Optimal Stopping Values and Prophet Inequalities for Negatively Dependent Random Variables

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Let $\mathbf{Y} = (Y_1, \cdots, Y_n)$ be random variables satisfying the weak negative dependence condition: $P(Y_i < a_i\mid Y_1 < a_1, \cdots, Y_{i-1}) \leq P(Y_i < a_i)$ for $i = 2, \cdots, n$ and all constants $a_1, \cdots, a_n$. Let $\mathbf{X} = (X_1, \cdots, X_n)$ have independent components, where $X_i$ and $Y_i$ have the same marginal distribution, $i = 1, \cdots, n$. It is shown that $V(\mathbf{X}) \leq V(\mathbf{Y})$, where $V(\mathbf{Y}) = \sup \{EY_t: t \text{is a stopping rule for} Y_1,\cdots, Y_n\}$. Also, the classical inequality which for nonnegative variables compares the expected return of a prophet $E\{Y_1 \vee \cdots \vee Y_n\}$ with that of the statistician $V(\mathbf{Y})$, i.e., $E\{Y_1 \vee \cdots \vee Y_n\} < 2V(\mathbf{Y})$, holds for nonnegative $\mathbf{Y}$ satisfying the negative dependence condition. Moreover, this inequality can be obtained by an explicitly described threshold rule $t(b)$, i.e., $E\{Y_1 \vee \cdots \vee Y_n\} < 2EY_{t(b)}$. Generalizations of this prophet inequality are given. Extensions of the results to infinite sequences are obtained.