I study how the shadow prices of a linear program that allocates an endowment of resources to customers behave as . I show the shadow prices (i) adhere to a concentration of measure, (ii) converge to a multivariate normal under central-limit-theorem scaling, and (iii) have a variance that decreases like . I use these results to prove that the expected regret in \cites{Li2019b} online linear program is , both when the customer variable distribution is known upfront and must be learned on the fly. I thus tighten \citeauthors{Li2019b} upper bound from to , and extend \cites{Lueker1995} lower bound to the multi-dimensional setting. I illustrate my new techniques with a simple analysis of \cites{Arlotto2019} multisecretary problem.
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