Source: Ann. Statist.
Volume 26, Number 3
All classical “prophet inequalities” for independent
random variables hold also in the case where only a noise-corrupted version of
those variables is observable. That is, if the pairs $(X_1, Z_1),\ldots,(X_n,
Z_n)$ are independent with arbitrary, known joint distributions, and only the
sequence $Z_1 ,\ldots,Z_n$ is observable, then all prophet inequalities which
would 1 n hold if the $X$’s were directly observable still hold, even
though the expected $X$-values (i.e., the payoffs) for both the prophet and
statistician, will be different. Our model includes, for example, the case when
$Z_i=X_i + Y_i$, where the $Y$’s are any sequence of independent random
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