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.

翻译：我研究了将 $n\beta \in \mathbb{R}^{m}$ 资源分配给 $n$ 个顾客的线性规划的对偶价格（影子价格）在 $n \rightarrow \infty$ 时的行为。结果表明，影子价格（i）满足测度集中性，（ii）在中心极限定理尺度下收敛于多元正态分布，且（iii）其方差以 $\Theta(1/n)$ 速率衰减。基于这些结论，我证明了在 \cites{Li2019b} 的在线线性规划问题中，无论顾客变量分布是预先已知还是需在线学习，期望遗憾均为 $\Theta(\log n)$。由此，我将 \citeauthors{Li2019b} 的上界从 $O(\log n \log \log n)$ 收紧至 $O(\log n)$，并将 \cites{Lueker1995} 的 $\Omega(\log n)$ 下界推广至多维场景。最后，我通过简化分析 \cites{Arlotto2019} 的多秘书问题，阐释了所提新技术的应用。