In this paper, we analyze the asymptotic behavior of the main characteristics of the mean-variance efficient frontier employing random matrix theory. Our particular interest covers the case when the dimension $p$ and the sample size $n$ tend to infinity simultaneously and their ratio $p/n$ tends to a positive constant $c\in(0,1)$. We neither impose any distributional nor structural assumptions on the asset returns. For the developed theoretical framework, some regularity conditions, like the existence of the $4$th moments, are needed. It is shown that two out of three quantities of interest are biased and overestimated by their sample counterparts under the high-dimensional asymptotic regime. This becomes evident based on the asymptotic deterministic equivalents of the sample plug-in estimators. Using them we construct consistent estimators of the three characteristics of the efficient frontier. It it shown that the additive and/or the multiplicative biases of the sample estimates are solely functions of the concentration ratio $c$. Furthermore, the asymptotic normality of the considered estimators of the parameters of the efficient frontier is proved. Verifying the theoretical results based on an extensive simulation study we show that the proposed estimator for the efficient frontier is a valuable alternative to the sample estimator for high dimensional data. Finally, we present an empirical application, where we estimate the efficient frontier based on the stocks included in S\&P 500 index.

翻译：本文运用随机矩阵理论分析了均值-方差有效前沿主要特征的渐近性质。我们特别关注维度$p$与样本量$n$同时趋于无穷且其比值$p/n$趋于正常数$c\in(0,1)$的情形。我们既未对资产收益施加任何分布假设，也未设定任何结构假设。在所建立的理论框架中，需要满足某些正则条件，例如四阶矩的存在性。研究表明，在高维渐近体系下，三个关注量中有两个存在偏差，且其样本对应量会高估真实值。这一结论基于样本插件估计量的渐近确定性等价形式得以明确。利用这些等价形式，我们构建了有效前沿三个特征的一致性估计量。研究证明样本估计量的加性和/或乘性偏差仅为集中率$c$的函数。此外，本文证明了所考虑的有效前沿参数估计量的渐近正态性。通过基于广泛模拟研究的理论结果验证，我们表明所提出的有效前沿估计量是高维数据样本估计量的有效替代方案。最后，我们给出了一个实证应用，基于标普500指数成分股对有效前沿进行估计。