High-dimensionality in statistics and portfolio optimization

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Many challenges in multivariate analysis arise when dealing with samples where the dimension is comparable to the sample size. This high-dimensional context can lead to inconsistencies or degenerated distributions of certain estimators, particularly those based on the sample covariance matrix, as their eigenvalues behave differently than those of the population covariance matrix. Notably, estimators for scatter also show interesting behavior in classical settings when the sample size significantly exceeds the dimension. A key contribution of this thesis is the establishment of the semicircle law for Tyler's M-estimator for scatter. It demonstrates that the empirical distribution of the eigenvalues of this estimator, when appropriately standardized, converges in probability to the semicircle law under spherical sampling, provided the sample dimension and size tend to infinity while their ratio approaches zero. The thesis also introduces a novel test for a scalar multiple of the covariance matrix of a normal population in high-dimensional settings. This test, inspired by the semicircle law in free probability theory, exhibits substantial local power when the dimension-to-sample size ratio is small. Additionally, it investigates the consistency and asymptotic distribution of standard estimators for the variance and mean of various portfolio returns under high-dimensionality, including the Sharpe ratios and weights of the global m