For k-fold matroid unions, there is a prophet inequality with competitive ratio 1-O(sqrt(log k/k)), proved via a new bicriterion concentration inequality for monotone 1-Lipschitz functions.
A sharp concentration inequality with applications
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A Bicriterion Concentration Inequality and Prophet Inequalities for $k$-Fold Matroid Unions
For k-fold matroid unions, there is a prophet inequality with competitive ratio 1-O(sqrt(log k/k)), proved via a new bicriterion concentration inequality for monotone 1-Lipschitz functions.