Stock Market of Random Matrix Theory to High-Dimensional Statistics

Abstract

This dissertation investigates the application of Random Matrix Theory (RMT) to the analysis of stock market behavior in high-dimensional statistical settings. As financial markets generate increasingly large and complex datasets, traditional statistical tools often fail to capture the true underlying correlations between assets, especially when the number of variables exceeds the number of observations common scenario in modern finance. RMT offers a robust mathematical framework for distinguishing genuine information from random noise in large correlation matrices. By analyzing the eigenvalue spectrum of empirical correlation matrices derived from asset return data, this study identifies deviations from the theoretical predictions of RMT, which often correspond to meaningful market signals or latent factors. The research explores both theoretical and empirical dimensions. On the theoretical side, it examines the implications of the Mar?enko–Pastur law and the behavior of eigenvalues in finite samples. On the empirical side, RMT-based filtering techniques apply to real-world financial datasets to enhance portfolio optimization, reduce estimation risk, and improve the stability of financial models under high-dimensional constraints. The findings demonstrate that RMT, when integrated with modern statistical learning techniques, provides a powerful approach for financial modeling, especially in contexts involving large asset universes and limited time series data. This study contributes to the growing body of literature that positions RMT as a cornerstone methodology in high-dimensional finance and paves the way for further interdisciplinary research at the intersection of statistical physics, econometrics, and machine learning