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\"{o}lder-$\beta$ local stationarity and $\alpha$-mixing, we establish a four-term coverage-error decomposition and derive the optimal calibration window $m^{\star} \asymp T^{2\beta/(2\beta+1)}$ with coverage-error rate $O(T^{-\beta/(2\beta+1)})$. A Le Cam two-point construction shows this rate is minimax-optimal over the H\"{o}lder-$\beta$ model class. The Bahadur representation is proved under both $\alpha$-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 $\pm2\%$ 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.