Rolling-Origin Conformal Prediction under Local Stationarity and Weak Dependence

We propose and analyse rolling-origin conformal prediction for time-series forecasting. The method calibrates the conformal quantile against the m most recent pseudo-out-of-sample forecast errors, adapting to serial dependence, volatility clustering, and distributional drift that invalidate classical conformal guarantees. Under H"older-β local stationarity and α-mixing, we establish a four-term coverage-error decomposition and derive the optimal calibration window mstar asymp T2β/(2β+1) with coverage-error rate O(T-β/(2β+1)). A Le Cam two-point construction shows this rate is minimax-optimal over the H"older-β model class. The Bahadur representation is proved under both α-mixing and the physical-dependence framework of Wu (2005). An oracle inequality formalises Winkler cross-validation as an adaptive window selector; the required uniform concentration condition is established in an appendix. Validation on six real series and 93 M4 competition series confirms the theory: rolling-origin calibration outperforms full-history calibration in 86% of comparisons (median Winkler improvement 12.3%), maintains coverage within ±2% of the 90% target at short and medium horizons, and the cross-frequency log-log regression slope 0.614 (95% CI [0.424, 0.805]) is consistent with the theoretical 2/3 after controlling for frequency fixed effects.