We present a simple stochastic representation for the joint distribution of sample estimates of three scalar parameters and two vectors of portfolio weights that characterize the minimum-variance frontier. This stochastic representation is useful for sampling observations efficiently, deriving moments in closed-form, and studying the distribution and performance of many portfolio strategies that are functions of these five variables. We also present the asymptotic joint distributions of these five variables for both the standard regime and the high-dimensional regime. Both asymptotic distributions are simpler than the finite-sample one, and the one for the high-dimensional regime, i.e. when the number of assets and the sample size go together to infinity at a constant rate, reveals the high-dimensional properties of the considered estimators. Our results extend upon T. Bodnar, H. Dette, N. Parolya and E. Thorstén [Sampling distributions of optimal portfolio weights and characteristics in low and large dimensions, Random Matrices: Theory Appl. 11 (2022) 2250008].

我们提出了一个简单的随机表示，用于描述最小方差前沿的三个标量参数和两个投资组合权重向量的样本估计的联合分布。这种随机表示对于有效地对观察结果进行采样、导出封闭形式的矩以及研究作为这五个变量的函数的许多投资组合策略的分布和性能非常有用。我们还提出了标准状态和高维状态下这五个变量的渐近联合分布。两种渐近分布都比有限样本分布更简单，而高维分布则更简单，即当资产数量和样本大小以恒定速率趋于无穷大时，揭示了所考虑的高维属性估计器。我们的结果延伸到 T. Bodnar、H. Dette、N. Parolya 和 E. Thorstén [低维和大维中最优投资组合权重和特征的抽样分布，随机矩阵：理论应用。 11（2022）2250008]。