Concentration, Information, and Distributional Stability in High-Dimensional Portfolios: A Talagrand Stability Index Approach.

This paper investigates the stability of high-dimensional financial portfolios using concentration inequalities, information-theoretic measures, optimal transport metrics, and financial network analysis. Asset returns are generated under both multivariate Gaussian and multivariate Student-t distributions. Equal Weight and Regularized Minimum Variance portfolios are evaluated across alternative portfolio dimensions. The results show that increasing portfolio dimension reduces portfolio risk, tail probabilities, and risk estimation errors, indicating stronger concentration and higher stability in high-dimensional settings. Entropy and mutual information measures reveal improved diversification and weaker dependence structures as portfolio size increases. To assess distributional robustness, a novel Talagrand Stability Index (TSI), combining Wasserstein distance and Kullback-Leibler divergence, is introduced. The results show that TSI decreases with portfolio dimension. Heavy-tailed Student-t returns generate weaker concentration effects, stronger dependence structures, and lower distributional stability than Gaussian returns. Mutual information-based financial networks reveal sparse and moderately interconnected dependence structures. To illustrate the practical applicability of the proposed framework, an empirical application based on daily returns of ten large U.S. equities during 2020-2025 is conducted, showing that the Regularized Minimum Variance portfolio achieves a marginally lower TSI than the Equal Weight portfolio. Robustness checks reported further indicate that this advantage is modest and outcome-dependent rather than decisive.