Technical note
The two prices of dependence
A vanishing coverage tax, and a fixed sharpness rent.
The result
Temporal dependence charges split conformal two different prices. The coverage price is a tax that vanishes with the calibration size: at most \(\min_\tau\{\tau/(n+1)+2\beta(\tau)\}\) for a stationary \(\beta\)-mixing score process (Barber–Pananjady). The sharpness price is a rent that does not: the per-step log-score regret of the marginal predictive against the past-conditional oracle equals the entropy-rate gap \(\Delta H = H(S_0)-h = I(S_0;\text{past})\), a constant independent of \(n\), untouched by any re-levelling of the marginal shape. For Gaussian score processes the rent exponentiates into width: oracle intervals are narrower by exactly \(e^{-\Delta H}\) at matched coverage — \(\sqrt{1-\phi^2}\) for AR(1), so less than half the width at \(\phi=0.9\).
The asymmetry
Fix the dependence and grow \(n\): the tax is \(O(\log n/n)\) for geometrically \(\beta\)-mixing series such as the AR(1) example, and vanishing in general; the rent is \(-\tfrac12\log(1-\phi^2)\) at every \(n\). The two prices answer different questions — what the certificate loses because the calibration data are dependent, versus what the forecast loses because the calibration ignores the dependence. The time-series conformal literature has concentrated on driving the first price to zero; the second is the one a forecaster pays at every step, and no amount of calibration data reduces it. This resolves, for the single-shape family, the pricing question posed on the conditional-coverage page: it is the information-gap identity with the side information taken to be the past.
Limits
The comparison prices marginality against the oracle, not against a feasible estimator; the width form is Gaussian; and when scores are residuals of a fixed predictor, dependence the predictor already absorbs does not appear in \(\Delta H\). The direction of the asymmetry survives all three limits: the tax vanishes in \(n\) and the rent does not.