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

This study provides a comprehensive empirical investigation of long-range dependence (LRD) in financial markets and evaluates the ability of deep generative models to reproduce such temporal structures. Using daily data from three representative sectors--equity (S&P 500, DAX, Nikkei 225), commodities (Wheat, Corn, Soybeans), and energy (UNG, USO, XLE)--we examine the presence of LRD through three complementary approaches: rescaled range (R/S) analysis, detrended fluctuation analysis (DFA), and an ARFIMA--FIGARCH model with Student's $t$-distributed innovations. The empirical evidence suggests that while mean returns exhibit limited persistence, pronounced long memory is consistently observed in conditional volatility across most assets. Building on these findings, we assess whether Quant Generative Adversarial Networks (Quant GANs) can learn and reproduce these stylized temporal dependencies. Although the generated series successfully mimic heavy-tailed return distributions and certain aspects of volatility clustering, they generally fail to capture the magnitude and consistency of LRD observed in real data, particularly in volatility dynamics. These results highlight an important limitation of current deep generative architectures in modeling slow-decaying dependence structures and underscore the need for incorporating explicit long-memory mechanisms when synthetic financial data are intended for risk management or long-horizon forecasting applications.