ARFIMA Processes in a Random Environment with Copula-Structured Dynamics: Application to Extreme Rainfall in Chad

Abstract Climate change is profoundly altering precipitation regimes worldwide, with a marked increase in the frequency and intensity of extreme rainfall events. Chad, spanning from the Saharan desert to the humid Sudanian savanna, offers a unique natural laboratory for studying extreme rainfall across con- trasting climatic zones under non-stationary conditions. However, classical statistical models face three major limitations: they assume parameter stationarity, ignore long-memory properties, and treat spa- tial dependence statically. This article develops an innovative theoretical framework to address these limitations simultaneously. We introduce ARFIMA processes evolving in a Random Environment with Copula-Structured Dynamics (ARFIMA-REE), where model parameters follow stochastic laws described by Archimedean copulas. This double-layer approach captures long memory via the fractional differ- encing parameter d(ηt), structural non-stationarity via a latent random environment {ηt}, and complex dependencies via dynamic copulas Ct. We establish existence, uniqueness, conditional stationarity, and asymptotic normality of estimators. The framework is validated on extreme rainfall data from 23 regions of Chad (1981-2023). Results reveal: (i) significant long memory across all regions, highest in the Sahara (Tibesti: d = 0.467); (ii) heavy-tailed behavior (ξ > 0) confirming unbounded extremes; (iii) spatial dependence dominated by the Clayton copula (87.5%), indicating stronger drought synchronization; and (iv) predictive superiority of ARFIMA-REE with RMSE reductions of 18.7% (vs ARFIMA) and 31.4% (vs ARIMA). Projections for 2050 indicate a north-south dichotomy: slight drying in the Sahara, stability in the Sahel, and modest increases in the Sudanian zone. 2020 Mathematics Subject Classification: 62M10, 62H05, 60K37, 62P12, 86A08, 62G32