Time series I
The coverage games
“Does conformal prediction work for time series?” is not one question. Four different guarantees travel under that name, and they can only be compared by what they hold fixed, what they average over, and what they assume. This page sorts the games. The attempts to say something conditional about your next step, given your recent past, are on the second page.
The setting, throughout: one observed path Z1,…,Zn+1 with Zi = (Xi, Yi), and split conformal prediction wrapped around an arbitrary score — absolute residual, quantile score, anything. Every result below is score-agnostic, so conformalized quantile regression is always the special case, never a separate theory. What varies between the games is the meaning of the probability in “P(Yn+1 is covered) ≈ 1−α”.
Game 1: restore the symmetry
Chernozhukov, Wüthrich & Zhu (2018; 2021). Exchangeability of points is dead, so permute blocks instead: if the series is close to block-exchangeable, the permutation logic behind conformal p-values goes through approximately, exactly in the block-exchangeable limit. The guarantee is marginal coverage averaged over paths, degraded by an ergodicity/mixing error term. The cost is a modified method and the approximation error, which you do not observe.
Game 2: keep the method, price the deviation
Barber, Candès, Ramdas & Tibshirani (2023) → Oliveira, Orenstein, Ramos & Romano (2024) → Barber & Pananjady (2026). The cleanest game, and the one with the surprise ending. Run vanilla split conformal, completely unmodified, on a dependent series, and ask: how far below 1−α can marginal coverage fall?
The progression matters. Barber et al. (2023) bound the coverage gap by a total-variation term per swapped calibration point — the right tool for distribution drift, but the wrong deformation for a time series, where swapping single points can change the joint distribution a lot even under short-range dependence. Oliveira et al. (2024) use blocking and empirical-process concentration to get a loss of order √(tmix/n) for β-mixing series — the first no-modification guarantee with rates meaningful for mixing series, at the classical square-root rate. Barber & Pananjady (2026) then close the question with a new “switch coefficient” and get, for a pretrained score on a stationary series,
coverage ≥ 1 − α − minτ { τ/(n+1) + 2β(τ) },
linear in τ/n where the blocking argument gave the square root — the guarantee converges faster than you have any right to expect. The reason is that the proof needs no concentration at all: the (1−α)-quantile of n scores moves by only about τ/n when τ entries are deleted (order statistics are insertion/deletion stable), and one total-variation swap then makes the test point look like a calibration point. Read the minimand as a bargain struck at every lag τ: pay τ/n in quantile slippage to buy a decoupling gap of τ steps, costing 2β(τ) in dependence. The result is also tight (a stationary construction achieves a constant fraction of the bound) and two-sided: a matching ceiling shows dependence cannot silently buy you 99% when you asked for 90% either (for continuous scores). Predictors with memory L cost an extra L/n; a score trained on an initial block needs a second buffer between training and calibration.
Scale check, from their moving-average experiments: with nominal 90%, n = 120 calibration points and an MA(t) dependence range of t = 12 (so t/n = 0.1), the worst-case bound permits coverage as low as 80%, while the observed loss is about one percentage point — roughly α·t/n. Dependence is a coverage tax, the bound caps the tax at (dependence range)/(calibration size), and in benign cases the bill comes in an order of magnitude under the cap. Watch the tax being levied in the dependence-tax demo, and see the drift demo for the failure this bound does not cover.
Averaged over: paths. This is coverage as a property of the ensemble of histories the process could produce — not of the history you are standing in.
Game 3: assume the model learns the truth
Xu & Xie (2021; 2023) — EnbPI, SPCI. Add two assumptions: the errors are strongly mixing, and the wrapped predictor (or conditional-quantile estimator) is consistent. Then you can prove much more — asymptotic conditional coverage, and interval width converging to the conditional oracle. The plain reading: the guarantee flows from the consistency assumption, not from the conformal wrapper. It is classical asymptotic statistics presented as conformal prediction. If the black box is not consistent, the theorems are silent; and the inconsistent black box is the case distribution-free inference exists for. Useful methods; just file the certificate under “model quality”, not “distribution-free”.
Game 4: drop probability altogether
Gibbs & Candès (2021; 2024), Bhatnagar et al. (2023), conformal PID (2023). The online game changes the target: guarantee that the time-average of coverage indicators along your one path tends to 1−α, for any sequence, adversarial included (conformal PID assumes bounded scores). No distribution, no ensemble — and therefore no probability statement about any particular time step. The certificate is real but weak in a specific way: it can be satisfied by intervals that oscillate between empty and infinite, and the average conceals when the misses happen. Bellman conformal inference (Yang, Candès & Lei 2024) repairs the degenerate-width problem by making expected length the explicit objective, solved by dynamic programming subject to the long-run constraint — an optimizer, not a distributional guarantee, but the most direct attack within this game.
What all four games refuse to say
At prediction time you can see the recent past. Every certificate above is an average — over the ensemble of paths (games 1–3) or over time along one path (game 4). A 90% certificate of either kind is compatible with 99% coverage in calm regimes and 60% in storms, indefinitely, provided storms are rare enough — and the storm is visible while you are in it. That is the information gap, and it is not a corner case: volatility clustering is the defining feature of most series people care about. The value of the certificate to a decision-maker at time t is precisely the gap between marginal and past-conditional coverage, which is this site’s running theme again: the coverage–sharpness gap as the mutual information between the conformal rank and the side information you hold, here the recent past (Marginally Useful; the fan note shows the same dependence as the width of the realized-coverage fan, and the Feynman–Wigner note gives the runnable check). The worked example makes the practical half of the point: the conditional quality of adaptive conformal methods comes from the conditional model they wrap, not from the conformal layer.
So the state of play: the marginal question is settled, generously and two-sidedly — dependence costs at most (dependence range)/(calibration size) in coverage, a tax so small it explains why nobody noticed it empirically. The conditional question is where the open problems live: continue to page two. And on the empirical question of which forecasting methods actually win, this site defers to the live, continuously running skaters benchmarks; nothing here is a benchmark.