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Change detection with conformal martingales: new optimal constructions, and suboptimality of existing methods

Swapnaneel Bhattacharyya, Aaditya Ramdas

arXiv:2609.27179v1math.STstat.MEstat.ML

TL;DR

Distribution-free changepoint detection is challenging when observations may be complex and both pre- and post-change laws are unknown. The paper builds weighted, restarted conformal procedures and proves delay bounds and optimality results for PFA and ARL control, improving on existing conformal martingale delay rates.

  • Problem

    Existing conformal test martingales provide distribution-free validity, but their delay optimality is not guaranteed and their delays can be large.

    Source: §1.1 · p. 6, §2.2 · pp. 9–10

  • Method

    The paper aggregates locally restarted conformal mixture martingales with deterministic weights, choosing weight conditions for PFA or ARL validity.

    Source: §1 · p. 4

  • Results

    The procedures achieve delay bounds of order at most log T/D_KL(H||U) for PFA and log γ/D_KL(H||U) for ARL, versus existing CTM delays that can be linear in T or √γ.

    Source: §1.1 · p. 6

  • Takeaways & Limitations

    For a fixed simple pair in a late-changepoint, fixed-power PFA window setting, the paper establishes a matching lower bound and uniform oracle RCMM upper bound.

    Source: §1.1 · p. 6

    Existing conformal test martingales can have PFA delay linear in T and ARL delay of order √γ.

    Source: §1.1 · p. 6, §2.2 · p. 10

Abstract

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We study distribution-free sequential changepoint detection for independent observations with unknown and unrestricted pre- and post-change laws. We build on the conformal test martingales and associated e-detectors of Vovk(2021), which control the probability of false alarm (PFA) and the average run length (ARL) respectively. The majority of these works focus on validity, with statistical efficiency usually left for simulations. We develop a comprehensive theory of how conformal p-values behave under non-exchangeable data with a changepoint at an unknown time $T$. We use this to analyze the post-change growth and resulting detection delay of conformal martingale methods, and prove that the standard existing methods are suboptimal for PFA and ARL control, and can lead to delays that are $Ω(T)$ and $Ω(\sqrt{\text{ARL}})$ respectively. We propose different conformal e-processes and e-detectors that are provably minimax optimal, with delays $Θ(\log T)$ and $Θ(\log \text{ARL})$ respectively, and have much shorter delays in simulations.

1 Introduction

The paper develops distribution-free sequential change detection for complex observations when pre- and post-change laws are unknown. It replaces existing conformal martingale procedures that lack delay-optimality guarantees with restarted, weighted constructions and analyzes their validity, delay, and optimality.

1 Introduction

  • PFA and ARL are distinct false-alarm criteria, and no nontrivial procedure can satisfy both; the paper’s ARL procedure also limits early stopping probability.PFA controls the probability of ever raising a false alarm, whereas ARL controls expected run length under no change.Source: §1 · p. 2, §1.1 · p. 6
  • A mixture over betting densities lets the detector adapt across possible post-change laws, since a density effective for one alternative may perform poorly for another.Mixing preserves the martingale property and supports distribution-free PFA validity.Source: §1 · p. 3
  • Restarting a martingale at each candidate changepoint and aggregating local evidence with weights reduces the dilution caused by a long pre-change history.A single martingale begun at time one carries pre-change capital that post-change evidence must overcome.Source: §1 · p. 3
  • The same weighted statistics provide PFA validity when weights are summable and ARL validity when each weight is at most one.The total weight governs the PFA regime, while the largest weight governs the ARL regime.Source: §1 · p. 4, §1.1 · p. 6

1.1 Summary of contributions

  • With a fixed changepoint, universal supermartingales over a composite distribution class face an impossibility result, motivating optimality analysis as T grows..Source: §1.1 · p. 6
  • The proposed PFA- and ARL-controlled procedures achieve delay bounds of order at most log T/D_KL(H||U) and log γ/D_KL(H||U), respectively, while existing conformal test martingales can have delays linear in T or √γ.The paper proves first-order minimax optimality in both regimes under its stated restricted formulations.Source: §1 · p. 5, §1.1 · p. 6
  • The paper proves a matching lower bound and upper bound for PFA detection in a fixed-simple-pair, late-changepoint, fixed-power window setting.Its score data-processing result quantifies information loss from score transformations; the likelihood ratio retains all raw-data information.Source: §1 · p. 5, §1.1 · p. 6

1.2 Related Work

  • Related work includes rank-based, multivariate, kernel, conformal, and e-value methods; restart weights allocate evidence across candidate changepoints, unlike weights for covariate or concept shift.Kernel approaches can face computational costs or depend on tuning choices, while the cited shift-adaptive conformal weights modify calibration.Source: §1.2 · pp. 7–8

2 Background and Main Results

The section establishes the independent, unknown-law changepoint setting and explains why valid conformal martingales may nevertheless detect late changes slowly. It then introduces weighted, restarted conformal evidence with distribution-free PFA or ARL validity.

2.1 Preliminaries

  • The framework assumes independent observations with unknown pre- and post-change laws, while exchangeability yields uniform conformal p-values for martingale validity.It defines e-processes through expected stopped values and e-detectors through expected stopped values bounded by expected stopping time.Source: §2.1 · p. 9, §2.2 · p. 9

2.2 Change detection using conformal test martingale and limitations

  • The earlier CTM construction uses betting martingales on conformal p-values to obtain PFA validity and CUSUM or Shiryaev–Roberts procedures for ARL validity.The cited validity results do not provide an optimality guarantee for detection delay.Source: §2.2 · p. 10
  • Existing CTM procedures can have delays of at least order T under PFA control and order √γ under ARL control.The PFA delay bound is conditional on surviving to the changepoint; the ARL example establishes a delay of order √γ with probability tending to one.Source: §2.2 · pp. 10–11

2.3 Our method for distribution-free change detection

  • The proposed procedures achieve detection delays of order log T for PFA control and log γ for ARL control, improving on existing conformal martingale methods.Source: §2.2 · p. 11
  • The construction aggregates betting functions and runs martingale evidence from candidate start times, so each restarted statistic measures evidence accrued since its start.Aggregating betting functions accommodates different departures from uniform conformal p-values when the post-change law is unknown.Source: §2.3 · pp. 11–12
  • A martingale started at time one retains pre-change capital, so late changes require post-change evidence to overcome accumulated dilution.Mixing over betting functions addresses the unknown post-change law but not the unknown changepoint.Source: §2.3 · p. 12
  • The weighted construction provides PFA-valid e-processes or ARL-valid e-detectors, with the choice and bounds on weights determining the regime.The stated validity result assumes exchangeable no-change observations and independent uniform randomizers.Source: §2.3 · p. 13

3 Optimality properties

The paper studies detection delay and optimality for distribution-free changepoint detectors, and shows why fixed changepoints cannot support universal consistency. In a growing-changepoint regime, its procedures achieve logarithmic delay bounds and restricted minimax guarantees, with score choice determining how much information is retained.

3.1 An impossibility result

  • At a fixed changepoint, no distribution-free PFA-valid detector can have power one against every alternative in a convex composite class.The paper therefore studies growing changepoints, where the pre-change sample remains non-negligible.Source: §3.1 · p. 15

3.2 Upper bounds on detection delay

  • PFA-controlled procedures achieve delay of order log T, while ARL-controlled procedures achieve delay of order log γ under the stated weight and growth conditions.The bounds are first-order sharp and match the corresponding information-theoretic lower bounds.Source: §3.2 · p. 17

3.3 A restricted minimax guarantee for PFA

  • For late changepoints, the proposed PFA-valid procedure attains the first-order restricted minimax detection window for both randomized and nonrandomized stopping times.The comparison is over procedures satisfying the PFA constraint and the specified late-changepoint range.Source: §3.3 · p. 18

3.4 A restricted minimax guarantee for ARL

  • ARL minimaxity requires an additional linear bound on early false alarms; under these restrictions, the proposed procedure attains the lower bound.A deterministic component can also force eventual threshold crossing while preserving PFA and ARL calibrations, without changing first-order delay when its weight is small.Source: §3.4 · pp. 19–20

3.5 Optimal score transformation

  • Score transformation cannot increase the KL information available for detection, and the likelihood-ratio score attains equality when its distribution under the pre-change law is continuous.The information loss is zero precisely when the likelihood ratio is measurable with respect to the transformed score.Source: §3.5 · p. 20

4 Change detection using different betting classes

The paper examines how betting-class assumptions affect delay guarantees for conformal detectors. A single profitable bet can provide logarithmic delay rates without oracle knowledge, while broader uniform guarantees require different assumptions and incur a separation-dependent cost.

4.1 Detection under a directional betting class

  • Under a directional betting class, late-alarm probability decreases exponentially in the number of post-change observations, but the expected-delay guarantee is truncated at a window proportional to T.The truncation cannot be removed because tail control is unavailable once post-change observations are a non-negligible fraction of the calibration pool.Source: §4.1 · pp. 21, 23
  • With a directional betting class, suitably weighted PFA- and ARL-controlled procedures have logarithmic delay bounds, with the evidence rate I⋆ replacing the oracle KL rate.The order is unchanged without knowledge of the pre- and post-change laws, although the delay constant changes.Source: §4.1 · pp. 22–23
  • A deterministic null-bet component forces eventual threshold crossing for every data sequence while preserving calibration, at a constant additive cost in the effective boundary.Choosing a small component regularizes the stopping time without changing first-order delay.Source: §4.1 · p. 23

4.2 Detection under other betting classes

  • A betting family need not satisfy the oracle’s smoothness or richness conditions for every alternative: positive prior mass on one profitable density suffices for its corresponding delay guarantee.The weighted procedure adapts to any profitable density that receives positive prior mass.Source: §4.1 · p. 23, §4.2 · p. 24
  • A positive prior mass on one profitable betting density yields delay governed by its evidence rate and a penalty log(1/vf) for its prior mass.Distributing prior mass across more candidates reduces each atom’s mass and can lengthen delay, while allowing a more favorable bet to be available.Source: §4.2 · p. 24
  • For alternatives uniformly separated from uniformity by Δ, the nonparametric delay bound scales as {boundary + log(1/Δ)}/Δ^2, subject to being o(T).A local alternative shows the Δ^-2 dependence cannot be improved in order.Source: §4.2 · pp. 24–25

5 Change detection using conformal mixture martingale

The section analyzes how post-change conformal p-values drive CMM growth and delay. It shows that ordinary CMM delay depends on the changepoint, while a suitable finite monitoring horizon leaves its local delay scale unchanged.

5 Change detection using conformal mixture martingale

  • The section develops CMM growth bounds as a building block for the paper’s detection-delay analysis.The argument decomposes the empirical CDF into pre- and post-change components; pre-change fluctuations are of order Tn^-1/2, while the post-change component converges to H.Source: §5 · p. 25
  • When rn/√Tn→∞, post-change empirical-CDF separation dominates pre-change fluctuations, producing p-value departure of order rn/mn above exchangeable m^-1/2 noise.This departure makes the change detectable in principle, but does not alone bound detection delay.Source: §5 · p. 25
  • The CMM’s post-change log-capital grows at order rn^2/Tn, so crossing boundary log bn requires delay on the scale √(Tn log bn).The stated delay bound assumes log Tn=o(log bn) and log bn=o(Tn); at a fixed threshold, the delay remains of order √Tn.Source: §5 · pp. 25–26
  • At a fixed threshold, CMM crosses within any admissible window rn with probability tending to one, but unconditional correct-window probability is limited by pre-change false alarms.For Tn=Θ(n), the text gives rn=an√n log n as an eventual almost-sure window.Source: §5 · pp. 26–27
  • A deterministic monitoring horizon can grow faster than any polynomial if its logarithm is negligible relative to Tn^(1−2γ); such truncation does not change the local CMM delay scale.The horizon also makes the monitored stopping time finite by construction.Source: §5 · p. 27

6 Simulation study

The simulations show that restarting improves detection-delay scaling relative to ordinary CMM, while both methods remain below nominal PFA levels under the no-change model. The results are consistent with the paper’s logarithmic-delay and distribution-free validity guarantees.

6 Simulation study

  • The simulations generate raw observations, compute randomized conformal p-values sequentially, and supply them to CMM and RCMM using the same left-tail betting density.The restarted procedure uses a PFA-controlled maximum statistic with summable weights.Source: §6 · p. 27
  • Across tested nominal PFA levels, CMM and RCMM’s empirical false-alarm probabilities remain below their targets under the no-change model.The simulations use finite monitoring horizons; the prose reports 1,000 replications and Monte Carlo confidence intervals.Source: §6 · pp. 28–29, Figure 4 · p. 29
  • RCMM’s median capped delay divided by log T stays bounded as T grows, while Vovk’s method diverges; ordinary CMM also degrades, indicating that restarting—not mixture aggregation—drives the improvement.The comparison uses 50 streams per changepoint location; ordinary CMM reaches the cap in representative Table 2 settings, while RCMM detects before the cap with increasing probability.Source: §6 · pp. 29–30, Figure 5 · p. 30

7 Conclusion

The paper introduces restarted conformal martingales with delay guarantees matching first-order minimax rates under both PFA and ARL control. Restarting addresses pre-change-history dilution, and raw-data simulations show RCMM operating on the log T scale while ordinary CMM delays grow rapidly.

7 Conclusion

  • In raw-data simulations, RCMM remains stable on the log T delay scale, while ordinary CMM reaches the monitoring cap in representative Table 2 settings.RCMM's median delay is on the log T scale, whereas CMM does not detect before the cap in those settings.Source: Table 2 · p. 31, §7 · p. 31
  • Restarting a martingale at every time point and aggregating these local processes avoids dilution from accumulated pre-change history.The process retains distribution-free validity while aggregating martingales associated with a class of betting functions.Source: §7 · pp. 30–31
  • The weight sequence calibrates one statistic for both criteria: its total weight controls PFA, while its largest weight controls ARL.Intermediate weight choices interpolate between the two formulations.Source: §7 · p. 31
  • With suitable weights, PFA- and ARL-controlled delays are at most of order log T/DKL(H||U) and log γ/DKL(H||U), respectively, matching first-order minimax rates.The paper also establishes explicit late-alarm bounds with probabilities that vanish exponentially in the number of post-change observations.Source: §7 · p. 32

Appendices

The appendices provide regularity conditions and constructions for betting classes, an impossibility theorem, and omitted lemmas and proofs.

Appendices

  • Appendix A discusses regularity properties of betting classes and gives concrete examples satisfying the conditions.Source: Appendices · p. 35
  • Appendix B contains an impossibility theorem for divergence of universally valid test supermartingales, supporting the reduced-filtration choice.Source: Appendices · p. 35
  • Appendix C contains omitted lemmas and proofs of the paper's results.Source: Appendices · p. 35

A Regularity of the betting class

The appendix gives several regularity conditions and concrete betting-class constructions for converting post-change departures from uniform conformal p-values into mixture growth. These include oracle-density support, broad local mixture richness, and a weaker directional detectability condition.

A Regularity of the betting class

  • Local KL support requires the betting mixture to reproduce the oracle log-density up to a logarithmic penalty, whose coefficient is not universally the parameter dimension.For a one-parameter exponential-tilt example, the transformed post-change law has positive KL divergence from uniform.Source: A Regularity of the betting class · p. 35
  • Uniform local mixture richness requires enough mixture mass on bets exploiting any small, nonzero departure of empirical p-values from uniformity.The threshold betting family varies the threshold to target departures across [0, 1], with a separate parameter controlling bet direction and strength.Source: A Regularity of the betting class · p. 36
  • Directional local detectability requires one regular betting path whose average under the post-change p-value law differs from its uniform average.The condition controls the log-bet's first-order approximation and local prior mass so mixture averaging retains growth along that direction.Source: A Regularity of the betting class · p. 36
  • A countable exponential-tilt family over dense mean-zero Lipschitz directions satisfies directional detectability for any continuous non-uniform transformed post-change law.Source: A Regularity of the betting class · pp. 36–37

B Impossibility of divergence for universal nonparametric test supermartingales under canonical filtration

Under a composite distribution class containing both pre- and post-change laws, universal full-filtration test supermartingales cannot diverge after a fixed change. The appendix also contrasts this impossibility with settings where the laws are known and explains why the paper works in a reduced filtration.

B Impossibility of divergence for universal nonparametric test supermartingales under canonical filtration

  • When a composite class M contains both P0 and P1, a test supermartingale valid for every law in M cannot diverge after a fixed change; a level-1/α alarm has detection probability at most α.Source: B Impossibility of divergence for… · p. 37
  • The requirement that both laws belong to the composite class is crucial: with known P0 and P1 and finite positive KL divergence, a likelihood-ratio martingale grows under the alternative.Source: B Impossibility of divergence for… · p. 38
  • Even a fixed conformal product bet decays exponentially under the simple null; the universal full-filtration impossibility motivates using a reduced filtration.Source: B Impossibility of divergence for… · p. 38

C Omitted Lemmata and Proofs

This appendix collects omitted proofs and technical results supporting validity claims, detection-delay bounds, and minimax window-length results for raw and conformal procedures. It also establishes guarantees for several detector variants and characterizes relationships among conformal p-values, oracle scores, and score transformations.

Non-asymptotic delay bound

  • Under Theorem 3.4’s conditions, the delay has an explicit non-asymptotic tail bound, with delayed-alarm probability decreasing exponentially in the number of post-change observations.The proposition applies to a general weight sequence and bounds both stopping times.Source: C Omitted Lemmata and Proofs · p. 38

Local comparison lemma

  • The local comparison lemma shows the next r post-change conformal p-values are iid with CDF H; its approximation converges in probability when rn→∞ and rn=o(Tn), and almost surely under stronger growth conditions.The almost-sure conclusion additionally assumes Tn/log n→∞ and rn/log n→∞.Source: C Omitted Lemmata and Proofs · p. 39

Proof of Example 2.2

  • When rn=o(Tn), the CTM’s conditional delay exceeds rn with probability tending to one, so its delay is not sublinear in the changepoint time.The proof first rules out threshold crossing through a fixed positive fraction of n post-change observations, conditional on surviving to n.Source: C Omitted Lemmata and Proofs · p. 40

Proof of Example 2.3

  • For CUSUM and SR with T=⌊√γ⌋, the conditional probability of surviving at least r≈c*T further observations tends to one, yielding delay on the order of √γ.The argument establishes this for both stopping times, using their pathwise ordering and a late-alarm probability bound.Source: C Omitted Lemmata and Proofs · pp. 41–43

Proof of Example 2.4

  • For the CMM, no threshold crossing occurs through ⌊c*Tn⌋ post-change observations with conditional probability tending to one, even though rn may be as small as K log Tn.The conclusion holds for deterministic rn=o(Tn), including rn=K log Tn.Source: C Omitted Lemmata and Proofs · p. 44
  • The CMM-CUSUM proof likewise bounds late alarms over rn≤c*Tn when Tn=⌊√γn⌋, after showing each CUSUM crossing implies an SR crossing at that time or earlier.The proof compares conformal p-values with post-change observations and bounds late-alarm probability using Markov’s inequality and a union bound.Source: C Omitted Lemmata and Proofs · pp. 45, 47

C Omitted Lemmata and Proofs

  • When r_n/√γ_n→c* and r_n/log γ_n→∞, both CMM-SR and CMM-CUSUM have expected-delay lower bounds and late-alarm probabilities tending to one.The same conditions yield the logarithmic-scale conclusions for both procedures.Source: C Omitted Lemmata and Proofs · p. 49
  • The start-indexed likelihood products and their weighted aggregate are nonnegative martingales, so Ville’s inequality supplies false-alarm control for the resulting e-process.The proof also establishes the exact weighted stopping inequality used to show that normalized S(w) and Z(w) are e-detectors and satisfy the stated ARL bounds.Source: C Omitted Lemmata and Proofs · pp. 49, 50, 52
  • Choosing ε so that A(D_KL(H∥U)−ε)>1 makes the bound tend to infinity; the local term lies in both weighted statistics, yielding crossing with probability tending to one, while lim inf_n q_n,w(b_n)>0 yields conditional and correct-window conclusions.For normalized weights, q_n,w(1/α) ≥ 1−α; for unit weights, B_n,w(γ_n) = log γ_n and q_n,w(γ_n) ≥ 1−T_n/γ_n, with T_n = o(γ_n) giving the conditional and correct-window conclusions.Source: C Omitted Lemmata and Proofs · pp. 54–55
  • At r_N=⌈c log N⌉ with c>1/D, raw and conformal procedures attain matching minimax window-length bounds, with PFA at most ᾱ and eventual power above 1−δ.The proof establishes matching lower bounds and verifies the conformal crossing condition uniformly over candidate change times.Source: C Omitted Lemmata and Proofs · pp. 56–57
  • The proofs establish Lorden’s and Pollak’s lower bounds by deriving a common no-change block hazard and geometric-survival bound, then applying the resulting argument to each criterion.For Pollak’s criterion, the likelihood-ratio argument yields the same no-change block hazard as for Lorden’s bound.Source: C Omitted Lemmata and Proofs · pp. 57–58

Score transformation and KL divergence

  • The score transform’s KL divergence equals the original divergence exactly when the likelihood ratio L is measurable with respect to T; s* = L satisfies this when its P0-law is continuous.This follows from conditional Jensen, with equality precisely when L = E_P0(L | T) almost surely.Source: C Omitted Lemmata and Proofs · p. 61

Weighted detector guarantees

  • When r ≤ δ0Tn, both weighted statistics cross bn by time Tn + r; the conditional late-alarm result follows by dividing the unconditional late-alarm bound by P(τ > Tn).Source: C Omitted Lemmata and Proofs · p. 63

Asymptotic crossing times

  • For r_n = floor(A B_n), the weighted rules have the stated crossing-time and conditional conclusions, with an almost-sure crossing result under the additional summability conditions.The proof chooses A > 1/I⋆ and uses the survival probability to obtain the conditional and correct-window conclusions.Source: C Omitted Lemmata and Proofs · p. 63

Weight-specific corollaries

  • For the stated weights, B_n,w(1/α) = log(1/α) + (1 + η) log T_n + O(1), while the unit-weight result uses B_n,w(γ_n) = log γ_n.The corresponding corollaries derive crossing-time conclusions and survival-based correct-window and conditional conclusions under their stated conditions.Source: C Omitted Lemmata and Proofs · pp. 63–64

Augmented mixture guarantees

  • The augmented betting mixture remains a nonnegative martingale with the same PFA and ARL bounds, and its effective delay boundary increases by log 1/(1 − ρ).For unit weights, the proof also gives a deterministic stopping bound.Source: C Omitted Lemmata and Proofs · pp. 64–65

Atomic construction

  • For the atomic construction, r_n = floor(A C_n,w,f(b_n)) gives the crossing-time result when A(I_f(H) − ε) > 1.The proof obtains the survival, conditional, and correct-window consequences from the corresponding probability bound.Source: C Omitted Lemmata and Proofs · p. 65

Fixed-row bound

  • Under Assumption 3, both weighted statistics stop by T_n + r outside the stated exceptional event.The proof of Theorem 4.12 supplies this bound for the fixed row.Source: C Omitted Lemmata and Proofs · p. 66

Rank-based procedures

  • The rank approximations converge uniformly in probability, and the stated crossing results under Assumptions 3 and 4 hold in probability and, under condition (36), eventually almost surely.The proofs use DKW bounds and rank comparisons; the additional condition makes the relevant deviation bounds summable.Source: C Omitted Lemmata and Proofs · pp. 66–68

Proof of Corollary 5.5

  • Under the stated growth inequality, Corollary 5.5 gives a lower bound for a correct post-change alarm and an almost-sure crossing-time conclusion; rn = o(Tn) ensures conditions (35) and (36).The lower bound is obtained by subtracting the pre-change alarm event.Source: C Omitted Lemmata and Proofs · p. 71

Proof of Proposition A.2

  • A nonzero empirical-CDF discrepancy yields an interval and betting-direction construction that verifies Assumption 3 for the proposed betting densities.The proof obtains an interval of length δ/4 with a signed discrepancy of at least δ/2, then a corresponding parameter interval.Source: C Omitted Lemmata and Proofs · p. 73

Proof of Proposition A.3

  • For any post-change score law H different from uniform, Proposition A.3 constructs a continuous direction and local prior mass satisfying Assumption 4.The constructed path has a nonuniform density, yielding positive KL divergence from uniform.Source: C Omitted Lemmata and Proofs · pp. 73–74

Proof of Theorem B.1

  • The constructed process is a test supermartingale under every choice of pre-change law, post-change law, and changepoint time.Its one-step conditional expectation is bounded by the preceding process both before and after the changepoint.Source: C Omitted Lemmata and Proofs · p. 75

Proof of Proposition B.2

  • After a fixed changepoint, the average log betting score converges almost surely to the uniform integral of log f, strictly below log of the integral of f when f is nonconstant.The proof compares post-change conformal p-values to randomized ranks within the post-change block, whose values are iid uniform.Source: C Omitted Lemmata and Proofs · p. 75

Proof of Lemma C.2

  • When rn = o(Tn), the conformal p-values over the post-change window approximate oracle scores in probability; stronger log-rate conditions yield almost-sure assertions.The proof compares ranks with transformed post-change observations and applies DKW, the weak law, and Borel–Cantelli.Source: C Omitted Lemmata and Proofs · p. 76

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