In many high-dimensional problems, the final 'summary statistic' is amenable to a univariate CLT, but with a complicated unknown covariance structure due to underlying dependencies. To take up a general perspective, we consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry-Esseen bound of optimal order $n^{-1/2}$ for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and $L^p$) distance. The results show that, in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate $n^{-1/2}$ cannot be reached, refuting a popular belief in the literature. This can be salvaged by simple methods: We reveal that in order to maintain the optimal speed of convergence $n^{-1/2}$, simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of Garch models of any order, functionals of dynamical systems arising from SDEs, iterated random functions and many more.

Many fields of modern sciences are faced with high-dimensional data sets. In this talk, we investigate the spectral properties of a large sample correlation matrix R. Results for the spectral distribution, extreme eigenvalues and functionals of the eigenvalues of R are presented in both light- and heavy-tailed cases. The findings are applied to independence testing and to the volume of random simplices.

This paper focuses on optimizing high-dimensional problems with linear constraints under uncertain conditions, often caused by noisy data. We address challenges in large-scale, data-driven applications where the parameter matrix is only approximately known due to noise and limited data samples. Our approach is a linear shrinkage method that blends random matrix theory and robust optimization principles. It aims to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables. This data-driven method excels in simulations, showing superior noise resilience and more stable performance in both achieving objectives and adhering to constraints compared to traditional robust optimization. Our findings highlight the effectiveness of our method in improving the robustness and reliability of optimization in high-dimensional, data-driven scenarios.

The paper "A Test on the Location of Tangency Portfolio for Small Sample Size and Singular Covariance Matrix" explores the tangency portfolio's positioning within feasible portfolios under specific conditions: small sample sizes and singular covariance matrices for asset returns. A new test is presented to determine this location, deriving the exact distribution of the test statistic under both null and alternative hypotheses and the high-dimensional asymptotic distribution as both portfolio dimension and sample size increase. The numerical study compares the asymptotic test with an exact finite sample test, showing effective performance. This research is pivotal for understanding tangency portfolio characteristics in challenging conditions, like singularity and limited data, enhancing financial decision-making strategies.