Before revision: Consider a gambler who observes the realizations of n independent, non-negative, distribution-labeled random variables arriving in a uniform random order and can stop the sequence at any time to obtain a reward of the most recent observation. In 2017, Correa et al. showed that when all distributions are identical, it is possible to design a stopping time that achieves a \beta \approx 0.745 fraction of the maximum value (the ``prophet'' benchmark), matching an upper bound of Hill and Kertz. In 2019, Correa et al. showed that when the distributions differ, it is no longer possible to achieve this bound: they prove an upper bound of \sqrt{3 - 1 < 0.74 on the best approximation ratio.
After revision: We introduce a new decomposition technique for random variables that maps a generic instance of the prophet inequalities problem to a new instance where all but a constant number of variables have a tractable structure that we refer to as (\varepsilon, \delta)-smallness. Using this technique, we make progress on several outstanding problems in the area: - 1 < 0.74 on the best approximation ratio.