  I study how the shadow prices of a linear program that allocates an endowment
of $n\beta \in \mathbb{R}^{m}$ resources to $n$ customers behave as $n
\rightarrow \infty$. 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 $\Theta(1/n)$. I use
these results to prove that the expected regret in \cites{Li2019b} online
linear program is $\Theta(\log n)$, both when the customer variable
distribution is known upfront and must be learned on the fly. I thus tighten
\citeauthors{Li2019b} upper bound from $O(\log n \log \log n)$ to $O(\log n)$,
and extend \cites{Lueker1995} $\Omega(\log n)$ lower bound to the
multi-dimensional setting. I illustrate my new techniques with a simple
analysis of \cites{Arlotto2019} multisecretary problem.
