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When fast mixing justifies pooled residuals

An exact finite-chain horizon-to-switching crossover for conditional coverage, alongside finite-calibration, irregular-grid, burn-in and transient-panel bounds.

Research note, 23 September 2026; extended 8 October 2026. A finite-state benchmark for homogenization and residual pooling; it does not assert distribution-free conditional coverage. This page is a summary; the proofs and the further results are in the full note.

What is being pooled?

Let $Y_t$ be an irreducible finite-state continuous-time chain with generator $mQ_0$ and invariant law $\pi$; the initial law need not equal $\pi$. A score $R_t$ (for example an absolute prediction residual) has continuous conditional CDF $F_i(r)$ when $Y_t=i$.

Emissions are independent given the states; the prediction rule and the emission laws are fixed. To forecast a terminal score at horizon $\Delta$ we pool calibration scores and choose the population threshold $q$ satisfying

The distinction is between coverage averaged over the stationary current state and coverage conditional on a specified current state. The forecast horizon $\Delta$ is also separate from the calibration spacing $h$: fast mixing along the former does not imply a large effective calibration sample along the latter.

The horizon sets the conditional error

Write $F(r)=(F_1(r),\ldots,F_d(r))^\top$ and $P_t=e^{mQ_0t}$. For any initial regime prior $\nu$, the conditional coverage error at the pooled population threshold is exactly

At $\Delta=c/m$ this is $(\nu e^{cQ_0}-\pi)F(q)$, independent of $m$ and generally nonzero; at fixed positive $\Delta$ it vanishes as $m\to\infty$. For the symmetric two-state chain with $m=1/\varepsilon$ the identity reduces to

Only for $\Delta/\varepsilon\to\infty$ does pooling become conditionally accurate in this benchmark. For $\Delta/\varepsilon\to0$, conditioning retains the full regime-specific difference.

The identity has a rate bound that does not need reversibility. Let $Q_0^*$ be the $L^2(\pi)$ adjoint, let $Q_s=(Q_0+Q_0^*)/2$, and let $\gamma_s$ be the smallest positive eigenvalue of $-Q_s$ on centered functions. Then

For a specified state $i$, $\|\delta_i/\pi-1\|_{2,\pi}=\sqrt{1/\pi_i-1}$ and the final score-CDF norm is at most $1/2$. The exponent is sharp among bounds with prefactor one. The route follows the nonreversible $L^2$ framework of Lawler and Sokal (1988).

Marginal validity is not conditional validity

If $Y_0$ is stationary, the two-state formula averages to exactly $1-\alpha$ at the population threshold, for every $\Delta$. This marginal identity says nothing about conditional coverage.

There is also a dependence-free obstruction. If an event $A$ has probability $p>0$ and a predictor has marginal miscoverage at most $\alpha$, then only

follows without additional structure, and the bound is sharp. An i.i.d. split-conformal example with a rare group nearly realizes the zero branch. Thus faster regime mixing, thinning and effective-sample-size corrections cannot by themselves turn marginal conformal validity into subgroup-conditional validity.

This is the elementary finite-sample counterpart of the general impossibility results of Barber, Candès, Ramdas and Tibshirani (2021).

The spacing sets the calibration variance

The population threshold is not what split conformal actually computes. Suppose $n$ calibration scores are sampled from the stationary process at spacing $h>0$, and let $\widehat F_n(r)=n^{-1}\sum_{\ell=1}^n {\bf1}\{R_{\ell h}\le r\}$ with $p_r=\pi F(r)$. With $a=e^{-m\gamma_sh}$, the exact finite-panel envelope is

The envelope is sharp: a symmetric two-state chain with conditional CDF values $(0,1)$ at the threshold attains it. When $m\gamma_sh\ll1$, adjacent scores carry nearly the same regime information, so $n$ can badly overstate the effective sample size.

On any deterministic grid $0=t_1<t_2<\cdots<t_n$ the factor becomes

and replacing an uneven grid by its average adjacent spacing is not a valid upper bound. For weights $w_i\geq0$, $\sum_iw_i=1$ and $R_{ij}=e^{-m\gamma_s|t_i-t_j|}$, the weighted empirical CDF satisfies $\operatorname{Var}\widehat F_w(r)\leq p_r(1-p_r)\,w^\top Rw$.

The unique minimizer is the generalized least-squares weight $w^\star=R^{-1}{\bf1}/({\bf1}^\top R^{-1}{\bf1})$, whose components are all positive. With the endpoint times fixed and the interior times free, equal spacing is the unique optimum.

A finite-sample statement

Let $\widehat q$ be the ordinary split-conformal order statistic of rank $k=\lceil(n+1)(1-\alpha)\rceil$, assuming $k\le n$ and no score ties. Select an integer $M\ge2$ and failure probability $\delta\in(0,1)$, let $V=V_n(a)$ or $V=V_{\boldsymbol t}$, and put $\eta_{n,M,\delta}=\frac1M+\sqrt{\frac{(M-1)V}{4n\delta}}$.

With probability at least $1-\delta$ over the stationary calibration sequence, for each possible observed state $Y_0=i$,

This bound separates calibration uncertainty from the irreducible horizon effect; it is intentionally loose, particularly its Chebyshev factor. If the chain starts from $\nu$ with a burn-in $b\ge0$, the confidence becomes $1-\delta-\zeta_b$ with $\zeta_b=\frac12e^{-m\gamma_sb}\left\|\frac{\nu}{\pi}-1\right\|_{L^2(\pi)}$.

Headline numbers

What it does not claim

The identity and additive-reversibilization bounds cover any irreducible finite-state CTMC and any deterministic regular or irregular observation grid. Scores are independent terminal emissions conditional on the terminal state, with fixed laws and a frozen prediction rule.

The variance and design results concern the marginal variance of an empirical calibration CDF. A variance-equivalent sample size is not a replacement for $n$ in a conformal rank law, and the minimum-variance weights do not make an ordinary weighted quantile a valid conformal rank rule.

Path functionals, predictors with memory, and overlapping path-score calibration require different formulas. No result here asserts distribution-free conditional coverage; the i.i.d. construction above shows why such a claim would require assumptions beyond fast mixing.

The full note also covers the joint panel-size and switching-rate limit, observation budgets and costs, a one-sided Cantelli threshold guarantee, the exact finite-state count law, total-variation comparisons with an i.i.d. panel, the large-panel exponential rate with its prefactor and corrections, and the start-up bias and variance of the empirical CDF.

Full note: conformal-crossover-full.html. Certificate: verify_crossover.py.