Long-Range Dependence in Financial Markets: Empirical Evidence and Generative Modeling Challenges

This study presents a comprehensive empirical investigation of the presence of long-range dependence (LRD) in the dynamics of major U.S. stock market indexes--S\&P 500, Dow Jones, and Nasdaq--at daily, weekly, and monthly frequencies. We employ three distinct methods: the classical rescaled range (R/S) analysis, the more robust detrended fluctuation analysis (DFA), and a sophisticated ARFIMA--FIGARCH model with Student's $t$-distributed innovations. Our results confirm the presence of LRD, primarily driven by long memory in volatility rather than in the mean returns. Building on these findings, we explore the capability of a modern deep learning approach, Quant generative adversarial networks (GANs), to learn and replicate the LRD observed in the empirical data. While Quant GANs effectively capture heavy-tailed distributions and some aspects of volatility clustering, they suffer from significant limitations in reproducing the LRD, particularly at higher frequencies. This work highlights the challenges and opportunities in using data-driven models for generating realistic financial time series that preserve complex temporal dependencies.