Pith. sign in

REVIEW 3 major objections 5 minor 73 references

Confidence Horizons

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper introduces confidence horizons: confidence sequences that are sharp on a bounded time window [m, Δm], recovering fixed-sample intervals at Δ=1 and infinite-horizon sequences as Δ→∞, with exact asymptotic coverage for means under

desk verdict Confidence horizons are a genuinely new, well-argued object — send to review, but verify the imported KMT coupling and soften the 'closed-form' claim. read the letter →

arxiv 2608.03889 v1 pith:6LGXV2WW submitted 2026-08-04 stat.ME math.STstat.MLstat.TH

Chase Mathis , Ian Waudby-Smith This is my paper
classification stat.MEmath.STstat.MLstat.TH MSC 62L1262L1062E2060F1760J65
keywords confidencehorizonanytime-validinferencesequencegroupsequentialmethodsstrongGaussianapproximationWienerprocessboundarycrossingmartingaledependenceadaptiveexperiments
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether giving up validity beyond a finite horizon buys sharper sequential inference, and answers yes. It defines an asymptotic confidence horizon: intervals around the running mean that cover the mean at every time in $[m, \Delta m]$ with limiting probability exactly $1-\alpha$, interpolating between a single CLT interval and an infinite-horizon confidence sequence. The width uses a boundary shape q and a quantile of $\sup_{s \in [1,\Delta]} |W(s)| s^{q-1}$, for which closed forms are given at q=0, 1/2, 1. The proof couples partial sums to a Wiener process via strong approximation and requires only log-rate variance estimation. Applied to adaptive Neyman allocation, the intervals are tight enough to stop early in simulations, and they turn group sequential designs into uniformly valid continuous analogues.

What carries the argument

The central object is the scaled Wiener supremum $\zeta(\Delta,q)=\sup_{s \in [1,\Delta]} |W(s)| s^{q-1}$ and its distribution $\Psi(\cdot;\Delta,q)$, whose quantile fixes the constant in front of the boundary. The argument converts the discrete-time partial-sum process into this Wiener process using distribution-uniform strong Gaussian approximation (KMT-type for independent data, Strassen’s theorem for martingales), then uses equicontinuity of $\Psi$ to make the approximation uniform in $\Delta$, $q$, and $P$. For $q \in \{0,1/2,1\}$, $\Psi$ and its one-sided version are expressed in closed form: bivariate normal integrals for $q=0,1$, and parabolic-cylinder/Ornstein–Uhlenbeck series for $q=1/2$.

What would settle it

Simulate many distributions from the stated class, including one with a moment $\kappa$ only slightly above 2, and compute worst-case coverage of the 0.95 horizon over $\Delta=3$ at $m=10^5$ using a variance estimator that satisfies Condition 3. If $\sup_P |\text{coverage} - 0.95|$ fails to shrink as $m$ grows, the uniform sharpness claim is false; a fixed-m experiment alone cannot falsify the asymptotic theorem.

Watch

Extended reading notes

Core claim

For independent data satisfying uniform moment and variance conditions, and for any variance estimator consistent at a log rate, the intervals $\hat{\mu}_t \pm \hat{\sigma}_t (m/t)^q m^{-1/2} \Psi^{-1}(1-\alpha; \Delta, q)$ form a sharp $(1-\alpha)$-asymptotic confidence horizon: $\lim_{m \to \infty} P(\forall t \in [m, \Delta m]: \mu_{P,t} \in \bar{C}_t) = 1-\alpha$, uniformly over distributions and $\Delta$ for $q \neq 1/2$, and weakly uniformly for $q=1/2$. The same guarantee holds for martingale-dependent data under almost-sure variance convergence and a Lindeberg-type uniform integrability condition. The construction is inverted to give p-values for bounded-horizon weak-null testing, and the same machinery upgrades the classical Wang–Tsiatis group sequential boundaries into

Load-bearing premise

Everything rests on the partial-sum process being uniformly close to a Wiener process with error $o_P(t^{-1/2} \log t)$ after studentization, together with a variance estimator that is consistent at a log rate; if the coupling rate or variance estimator fails, exact horizon coverage is not guaranteed.

Editorial extensions

If this is right

  • At Δ=1 the confidence horizon reduces to the usual CLT interval; as Δ→∞ it approaches an asymptotic confidence sequence, so the construction continuously interpolates between fixed-sample and infinite-horizon inference.
  • Analysts can peek at every time in a bounded window and stop early, getting exact asymptotic coverage instead of conservative infinite-horizon guarantees.
  • Quantiles for q∈{0,1/2,1} are computed by at most one numerical integration, bypassing repeated K-dimensional integration in group sequential designs.
  • Wang–Tsiatis, Pocock, and O’Brien–Fleming boundaries become continuous analogues and inherit uniform validity and martingale-dependence validity.
  • In adaptive experiments under Neyman allocation, horizons separate arms and allow earlier stopping that asymptotic confidence sequences cannot.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper’s power analysis suggests a design rule that is only partly developed: larger q gives more end-of-window power, while q=1/2 minimizes expected stopping time, so q should be chosen by the analyst’s objective.
  • The same mechanism—Gaussian coupling plus log-rate variance consistency—should extend to asymptotically linear semiparametric estimators beyond means, since only those two ingredients are used.
  • Multiple disjoint horizons could be combined without a union bound by simulating the supremum over a union of intervals, an appendix sketch that could become a practical tool for diffuse or bimodal prior signal locations.
  • The residual-type variance estimator recommendation for time-varying means suggests pairing confidence horizons with difference-based variance estimators in non-stationary streams.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces "asymptotic confidence horizons" (AsympCHs): intervals \bar C^{(m,\Delta)}_t = \hat\mu_t \pm \hat\sigma_t (m/t)^q m^{-1/2} \Psi^{-1}(1-\alpha;\Delta,q), designed to satisfy the bounded-horizon coverage guarantee lim_{m\to\infty} P(\forall t\in[m,\Delta m]: \mu_{P,t}\in\bar C_t) = 1-\alpha. The construction interpolates between fixed-sample CLT intervals (\Delta=1) and asymptotic confidence sequences (\Delta\to\infty). The main results are: Theorem 3.3 for independent, non-identically distributed data under Conditions 1-3; Theorem 3.7 for martingale-dependent data under Conditions 4-5 and Condition 3; one-sided analogues in Proposition 3.4; p-values for the weak null in Proposition 3.6; a Neyman-allocation application in Proposition 4.1; and uniformly valid group sequential analogues in Propositions 5.2 and 5.4. The proofs use strong Gaussian approximations (Lemma A.12, imported from [68], and Lemma A.13, via Strassen), followed by a continuous-time boundary-crossing analysis. Appendix B gives analytic expressions for the required quantiles for q\in\{0,1/2,1\}.

Significance. If the central result is correct, this is a useful contribution to asymptotic sequential inference. It quantifies the statistical gain from restricting validity to a finite horizon, yields explicit boundary-crossing quantiles, and provides distribution-uniform versions of both confidence horizons and group sequential methods. The paper is careful about the distinction between pointwise and uniform validity, and it is honest about finite-sample discrepancies in the Neyman-allocation simulation. The code links are a positive feature. However, the main theorem is built on an external distribution-uniform coupling whose hypotheses are not reproduced in the manuscript; the central claim is therefore conditional on an unexamined result.

major comments (3)
  1. [§A.2, Lemma A.12, and proof of Theorem 3.3 (§A.3.1)] The proof of Theorem 3.3 verifies Condition C2 (the additive Gaussian coupling at rate o_P(t^\gamma\log t), \gamma<-1/2) entirely by Lemma A.12, which states that [68, Corollary 3.4] implies such a coupling for sequences satisfying Conditions 1 and 2. The corollary is not stated, and its hypotheses are not checked against Conditions 1-2. This is load-bearing: Lemma A.14 replaces the empirical process by a Wiener process precisely at this step, and the sharp coverage claim of Theorem 3.3 does not follow if [68, Corollary 3.4] requires stronger or different assumptions (for example, larger moment order, i.i.d. data, or an exponential moment). The authors should state the corollary and either prove it under the stated conditions or add the required hypotheses to Theorem 3.3.
  2. [§3.2, Condition 3 and Variance Estimation paragraph] Theorem 3.3 is conditional on Condition 3, but the manuscript does not establish that the usual sample variance satisfies Condition 3 for independent non-identically distributed data under Conditions 1-2. The variance-estimation discussion admits that the raw sample variance may fail when the running mean varies (Eq. (13)) and recommends residual-type estimators. This is not itself an inconsistency, but the practical scope of Theorem 3.3 is narrower than the abstract's emphasis on i.i.d./independent means suggests. Please state more prominently which estimators are covered and under which additional conditions, especially in the non-i.i.d. case.
  3. [§5.2, Propositions 5.2 and 5.4] The claimed uniform validity of Wang-Tsiatis group sequential methods for independent and martingale-dependent data inherits the same unexamined dependency: Proposition 5.2 is proved by checking C1 and C2, and C2 is obtained from Lemma A.12 / [68, Corollary 3.4]. The discussion in Remark 5.3 emphasizes uniformity over K and over the design, which is a strong and interesting claim; the proof should make the supporting coupling theorem explicit so that the uniformity claim is verifiable.
minor comments (5)
  1. [Abstract and §B.2 (Remark 3.5)] The phrase "closed-form distribution functions" for q=1/2 is somewhat strong: Proposition B.4 and Lemma B.6 express the distribution through infinite series involving zeros of parabolic cylinder functions, so computing the quantile still requires root-finding. The paper should temper "closed-form" or define the sense in which these expressions are closed-form.
  2. [§3.2, paragraph after Theorem 3.3] For general q, the authors note that quantiles computed by Monte Carlo "may be anticonservative for any fixed level of precision." This is an important caveat for users; it would be helpful to add a recommendation about error control or interpolation between the closed-form cases.
  3. [§4, Fig. 5 and surrounding text] The Neyman-allocation simulation reports per-arm coverage estimates that fall slightly below 0.95 for some arms, and the text correctly attributes this to finite-sample behavior of an asymptotic guarantee. Reporting Monte Carlo standard errors would make the discrepancy easier to interpret.
  4. [§B.2, Lemma B.6] The notation B_\mu A_{\nu_n}(x) in Lemma B.6 is confusing: it denotes the derivative with respect to the index \mu of A_\mu(x) evaluated at \nu_n. Please clarify the notation, since this is a key quantity in the two-sided q=1/2 formula.
  5. [Throughout] There are occasional typos and stylistic issues (e.g., "V ariance Estimation" in the section heading, and some missing periods). A careful proofreading pass would help.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: quantiles are Wiener-process crossing values, not fitted; the main self-cited dependency (KMT approximation from [68]) is external and does not reduce the theorem to its inputs.

full rationale

The central derivation is self-contained in the relevant sense. The quantile Ψ^{-1}(1−α; Δ, q) is defined as the inverse of the distribution of sup_{s∈[1,Δ]} |W(s)| s^{q−1} for a standard Wiener process, and Section B derives closed forms for q∈{0,1/2,1} from external Wiener/Ornstein–Uhlenbeck hitting-time results. Theorem 3.3 is proved by showing, under Conditions 1–3, that the studentized running-mean process converges uniformly to this Wiener functional (Lemma A.14) and then inverting the limiting distribution. That is a standard strong-approximation/CLT argument, not a definitional equivalence or a fit renamed as a prediction. The only author-overlapping dependency is Lemma A.12, which imports Corollary 3.4 of [68] (Waudby-Smith, Larsson, Ramdas) to obtain a distribution-uniform KMT coupling. This is load-bearing for uniformity, but it is an external, separately stated theorem with its own assumptions (moment summability and variance stability, given here as Conditions 1–2) that do not include the confidence-horizon guarantee. Under the review rules, such a citation is independent support and does not by itself constitute circularity. The manuscript does not restate the full hypotheses of [68, Cor. 3.4], which is a completeness/correctness risk rather than a circularity. The Monte Carlo remark for general q explicitly warns that simulation-based quantiles may be anticonservative, which is an honest limitation, not a fitted-input maneuver. The connections to Pocock, O’Brien–Fleming, and Wang–Tsiatis are explicitly framed as connections and the paper derives its own closed-form distributions rather than renaming the known group-sequential constants. Overall, no step reduces by construction to its inputs; the score reflects only the presence of a load-bearing but non-circular self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data-fitted parameters: q, m, and Δ are user-selected design choices. No new physical or probabilistic entities are introduced; the confidence horizon is a new statistical object, not an entity in the graviton sense.

assumptions (5)
  • standard math Komlós-Major-Tusnády strong approximation holds uniformly over P at the stated rate (Lemma A.12, using [68, Cor. 3.4]).
    Used to couple the partial-sum process with a Wiener process; if the rate fails, the coverage uniformity proof collapses.
  • standard math Strassen's strong approximation for martingales holds under Condition 4 and 5 (Lemma A.13, [57]).
    Extends the coupling to martingale-dependent data.
  • standard math The distribution functions F and F_+ are equicontinuous and invertible on the needed domains (Lemma A.4, A.8).
    Needed for Pólya-type upgrade from pointwise to uniform convergence.
  • domain assumption Moment conditions Condition 1, 2, and variance estimator consistency Condition 3 for independent data; Conditions 4 and 5 for martingale data.
    These are the user-facing assumptions on the data and estimators; they define the class P over which uniformity holds.
  • domain assumption The target parameter is the running mean \tilde{μ}_{P,t}, not a fixed parameter, for non-identically distributed data.
    The CH covers the running mean; for i.i.d. or constant-conditional-mean martingales this equals the fixed parameter.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Confidence Horizons." pith.science (2026). https://pith.science/paper/6LGXV2WW

@misc{pith2026260803889,
  author       = {Pith},
  title        = {Pith review of: Confidence Horizons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LGXV2WW}},
  note         = {Machine review of arXiv:2608.03889}
}
read the original abstract

Anytime-valid inference enables analysts to continuously monitor their data and stop experiments early. However, the majority of these methods incur a certain conservativeness by remaining valid on infinite time horizons. In practice, a bound on the horizon may be imposed due to budgetary, practical, or ethical constraints. In this paper, we ask the question: "Is it possible to obtain sharper large-sample anytime-valid inference by forgoing validity beyond some finite time horizon?". We provide a positive answer to this question by proposing a family of statistical objects that we call "confidence horizons". These objects can be viewed as large-sample confidence sequences on bounded time horizons, or alternatively as group sequential repeated confidence intervals with a maximal number of interim peeking times. We make explicit connections to the group sequential boundaries of Pocock [1977], O'Brien--Fleming [1979], and Wang--Tsiatis [1987]. We derive closed-form distribution functions of certain statistics which can be used to calculate the asymptotic quantiles of confidence horizons exactly, sidestepping the repeated integration typically employed in group sequential methods. We illustrate the use of confidence horizons for treatment effect estimation in sequentially randomized experiments under adaptive Neyman allocation.

Figures

Figures reproduced from arXiv: 2608.03889 by the authors.

Figure 1
Figure 1. A comparison of widths of CLT-based confidence intervals, asymptotic confidence sequences [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. An illustration of how confidence horizons compare to other objects in the literature [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Example of possible confidence horizons illustrating their differences from confidence [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: pXtq 8 t“1 are independent but not identically distributed random variables. We let Xt “ µt ` εt where µt “ 5 sinp3{10π ? tq and εt i.i.d. „ Laplacep0, 1{ ? 2q. We see that using a residualized variance estimator like in (15) gives tighter bounds, suggesting that one s…
Figure 5
Figure 5. Figure 5: One realization of an adaptive experiment with 6 arms, each with different means and [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: We focus on the top two arms and compare AsympCHs against AsympCSs and [ [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Asymptotic confidence horizons form continuous analogues of common group sequential [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: For a Gaussian experiment, we consider the tradeoffs between power and expected stopping [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Comparing widths and coverage over a bounded window [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Comparing widths and coverage over a bounded window [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: Approximation error for AsympCH as m Ñ 8. For a skewed Beta distribution, asymptotics kick in around m “ 100. If the data are more symmetric, this approximation happens quite rapidly. The phenomenon connects with how the third moment influences the pointwise Berry–Ess…
Figure 12
Figure 12. Figure 12: Illustration of candidates and imposters. [PITH_FULL_IMAGE:figures/full_fig_p049_12.png]
Figure 13
Figure 13. Figure 13: Illustration of numerical calculation of group sequential quantiles as [PITH_FULL_IMAGE:figures/full_fig_p058_13.png]
Figure 14
Figure 14. Figure 14: Comparison of confidence horizons with the [PITH_FULL_IMAGE:figures/full_fig_p059_14.png]
Figure 15
Figure 15. Figure 15: Distributions of the J “ 6 arms. Notice that while arm 5 and arm 6 have similar means, arm 5 has a much larger spread compared to arm 6. Therefore, [35, Algorithm 1] should sample from arm 5 more than arm 6. In addition, the arms have different levels of skewness and …
Figure 16
Figure 16. Figure 16: The full Neyman allocation experiment across AsympCS, AsympCH: [PITH_FULL_IMAGE:figures/full_fig_p060_16.png]
Figure 17
Figure 17. Figure 17: Power analysis over m, ∆. As m increases the AsympCH is more powerful. Taking m Ñ 8 will yield a power 1 test. In addition, if ∆ increases, the AsympCH is more powerful. Power Stopping-time Tradeoff Power is one metric an experimenter can use to design an experiment. …
Figure 18
Figure 18. Figure 18: Power analysis for the AsympCH for different values of [PITH_FULL_IMAGE:figures/full_fig_p062_18.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

73 extracted references · 65 canonical work pages

  1. [68]

    Waudby-Smith, M

    I. Waudby-Smith, M. Larsson, and A. Ramdas. Nonasymptotic and distribution-uniform Koml´ os-Major-Tusn´ ady approximation.arXiv preprint arXiv:2502.06188, 2025. 35

    arXiv 2025

  2. [1]

    Agrawal and A

    S. Agrawal and A. Ramdas. On stopping times of power-one sequential tests: Tight lower and upper bounds.arXiv preprint arXiv:2504.19952, 2025. 61

    arXiv 2025

  3. [2]

    Alili, P

    L. Alili, P. Patie, and J. L. Pedersen. Representations of the First Hitting Time Density of an Ornstein-Uhlenbeck Process 1.Stochastic Models, 21(4):967–980, Oct. 2005. 10, 52, 53

    work page 2005

  4. [3]

    Anderson.gsDesign: Group Sequential Design, 2026

    K. Anderson.gsDesign: Group Sequential Design, 2026. URL https://keaven.github.io/ gsDesign/. R package version 3.11.0.9003. 18

    work page 2026

  5. [4]

    Armitage, C

    P. Armitage, C. McPherson, and B. Rowe. Repeated significance tests on accumulating data. Journal of the Royal Statistical Society: Series A (General), 132(2):235–244, 1969. 18

    work page 1969

  6. [5]

    R. R. Bahadur and L. J. Savage. The Nonexistence of Certain Statistical Procedures in Nonparametric Problems.The Annals of Mathematical Statistics, 27(4):1115–1122, Dec. 1956. 5

    work page 1956

  7. [6]

    Bellman and T

    R. Bellman and T. Harris. Recurrence times for the Ehrenfest model.Pacific Journal of Mathematics, 1951. 10

    work page 1951

  8. [7]

    A. C. Berry. The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society, 49(1):122–136, 1941. 23 24

    work page 1941

Show all 73 references
  1. [8]

    Besag and P

    J. Besag and P. Clifford. Sequential Monte Carlo p-values.Biometrika, 78(2):301–304, 1991. 15

    1991

  2. [9]

    Bibaut, N

    A. Bibaut, N. Kallus, and M. Lindon. Near-optimal non-parametric sequential tests and confidence sequences with possibly dependent observations.arXiv preprint arXiv:2212.14411,

    arXiv

  3. [10]

    Blanchet-Scalliet, D

    C. Blanchet-Scalliet, D. Dorobantu, and L. Gay. Joint law of an Ornstein–Uhlenbeck process and its supremum.Journal of Applied Probability, 57(2):541–558, 2020. 52

    2020

  4. [11]

    Blanchet-Scalliet, D

    C. Blanchet-Scalliet, D. Dorobantu, and B. Nieto. A pseudo-likelihood estimator of the Ornstein–Uhlenbeck parameters from suprema observations.Statistical Inference for Stochastic Processes, 27(2):407–425, 2024. 10, 51, 52

    2024

  5. [12]

    Blanchet-Scalliet, D

    C. Blanchet-Scalliet, D. Dorobantu, and B. Nieto. On the distribution of the first exit time from an interval of a threshold Ornstein-Uhlenbeck process. 2025. 10, 51, 54

    2025

  6. [13]

    V. I. Bogachev.Gaussian measures. Number 62. American Mathematical Soc., 1998. 47

    1998

  7. [14]

    L. Breiman. First exit times from a square root boundary. InProceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, Part 2, volume 5, pages 9–17. University of California Press, 1967. 10

    1967

  8. [15]

    Chugg, E

    B. Chugg, E. Gauthier, M. I. Jordan, A. Ramdas, and I. Waudby-Smith. Post-hoc large-sample statistical inference.arXiv preprint arXiv:2603.08002, 2026. 4

    2026

  9. [16]

    Clerico, T

    E. Clerico, T. Wegel, I. Azangulov, and P. Rebeschini. Time-sensitive anytime-valid testing. arXiv preprint arXiv:2605.06521, 2026. 4

    2026 arXiv

  10. [17]

    T. Cook, A. Mishler, and A. Ramdas. Semiparametric efficient inference in adaptive experiments. InCausal Learning and Reasoning, pages 1033–1064. PMLR, 2024. 13, 15

    2024

  11. [18]

    Csorgo and P

    M. Csorgo and P. Revesz. How Big are the Increments of a Wiener Process?The Annals of Probability, 7(4):731–737, Aug. 1979. 44

    1979

  12. [19]

    J. Dai, P. Gradu, and C. Harshaw. Clip-ogd: An experimental design for adaptive neyman allocation in sequential experiments.Advances in Neural Information Processing Systems, 36: 32235–32269, 2023. 13, 14

    2023

  13. [20]

    D. A. Darling and H. Robbins. Iterated logarithm inequalities.Proceedings of the National Academy of Sciences, 57(5):1188–1192, 1967. 61

    1967

  14. [21]

    D. A. Darling and A. J. F. Siegert. The First Passage Problem for a Continuous Markov Process.The Annals of Mathematical Statistics, 24(4):624–639, Dec. 1953. 10

    1953

  15. [22]

    D. M. De Long. Crossing probabilities for a square root boundary by a Bessel process. Communications in Statistics-Theory and Methods, 10(21):2197–2213, 1981. 10

    1981

  16. [23]

    J. L. Doob. The Brownian movement and stochastic equations.Annals of mathematics, 43(2): 351–369, 1942. 52

    1942

  17. [24]

    C.-G. Esseen. On the Liapunoff limit of error in the theory of probability, 1942. 23

    1942

  18. [25]

    Fischer and A

    L. Fischer and A. Ramdas. Sequential Monte Carlo testing by betting.Journal of the Royal Statistical Society Series B: Statistical Methodology, 87(4):1200–1220, 2025. 15 25

    2025

  19. [26]

    Hadad, D

    V. Hadad, D. A. Hirshberg, R. Zhan, S. Wager, and S. Athey. Confidence intervals for policy evaluation in adaptive experiments.Proceedings of the national academy of sciences, 118(15),

  20. [27]

    P. Hall, J. Kay, and D. Titterington. Asymptotically optimal difference-based estimation of variance in nonparametric regression.Biometrika, 77(3):521–528, 1990. 11

    1990

  21. [28]

    Hoeffding

    W. Hoeffding. Probability inequalities for sums of bounded random variables.Journal of the American statistical association, 58(301):13–30, 1963. 5

    1963

  22. [29]

    S. R. Howard, A. Ramdas, J. McAuliffe, and J. Sekhon. Time-uniform, nonparametric, nonasymptotic confidence sequences.The Annals of Statistics, 49(2):1055–1080, 2021. 5, 6, 22

    2021

  23. [30]

    Imbens, L

    G. Imbens, L. Masoero, A. Rakhlin, T. S. Richardson, and S. Vijaykumar. Demonstration Experiments.arXiv preprint arXiv:2603.06941, 2026. 9

    2026 arXiv

  24. [31]

    Jennison and B

    C. Jennison and B. W. Turnbull. Interim analyses: the repeated confidence interval approach. Journal of the Royal Statistical Society Series B: Statistical Methodology, 51(3):305–334, 1989. 4

    1989

  25. [32]

    Jennison and B

    C. Jennison and B. W. Turnbull. Distribution theory of group sequential t, χ2 and F-tests for general linear models.Sequential Analysis, 16(4):295–317, 1997. 4, 18

    1997

  26. [33]

    Jennison and B

    C. Jennison and B. W. Turnbull. Group-sequential analysis incorporating covariate information. Journal of the American Statistical Association, 92(440):1330–1341, 1997. 18

    1997

  27. [34]

    Jennison and B

    C. Jennison and B. W. Turnbull.Group Sequential Methods with Applications to Clinical Trials. Chapman and Hall/CRC, 2000. 3, 17, 18, 19

    2000

  28. [35]

    M. Kato, T. Ishihara, J. Honda, and Y. Narita. Efficient adaptive experimental design for average treatment effect estimation.arXiv preprint arXiv:2002.05308, 2020. 13, 14, 15, 16, 59

    2002 arXiv

  29. [36]

    Koml´ os, P

    J. Koml´ os, P. Major, and G. Tusn´ ady. An approximation of partial sums of independent R V’s, and the sample DF. II.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 34 (1):33–58, 1976. 35

    1976

  30. [37]

    N. W. Koning and S. Van Meer. Anytime validity is free: inducing sequential tests.Journal of the Royal Statistical Society Series B: Statistical Methodology, page qkag050, 2026. 4

    2026

  31. [38]

    Lamperti

    J. Lamperti. Semi-stable stochastic processes.Transactions of the American mathematical Society, 104(1):62–78, 1962. 52

    1962

  32. [39]

    K. G. Lan and D. L. DeMets. Discrete sequential boundaries for clinical trials.Biometrika, pages 659–663, 1983. 18, 59

    1983

  33. [40]

    Lattimore and C

    T. Lattimore and C. Szepesv´ ari.Bandit algorithms. Cambridge University Press, 2020. 14

    2020

  34. [41]

    Liang and I

    B. Liang and I. Bojinov. An experimental design for anytime-valid causal inference on multi- armed bandits.arXiv preprint arXiv:2311.05794, 2023. 14

    2023 arXiv

  35. [42]

    Marcinkiewicz and A

    J. Marcinkiewicz and A. Zygmund. Sur les fonctions ind´ ependantes.Fundamenta Mathematicae, 29:60–90, 1937. 12 26

    1937

  36. [43]

    Massiani, S

    P.-F. Massiani, S. Schulze, and M. Mollenhauer. Asymptotic e-processes.arXiv preprint arXiv:2604.19353, 2026. 4

    2026 arXiv

  37. [44]

    McDiarmid and others

    C. McDiarmid and others. On the method of bounded differences.Surveys in combinatorics, 141(1):148–188, 1989. 5

    1989

  38. [45]

    M¨ orters and Y

    P. M¨ orters and Y. Peres.Brownian motion, volume 30. Cambridge University Press, 2010. 44, 56

    2010

  39. [46]

    Neopane, A

    O. Neopane, A. Ramdas, and A. Singh. Logarithmic neyman regret for adaptive estimation of the average treatment effect.arXiv preprint arXiv:2411.14341, 2024. 13

    2024 arXiv

  40. [47]

    J. Neyman. On the Application of Probability Theory to Agricultural Experiments. Essay on Principles. Section 9 (translated, reprinted ed.).Statistical Science, 5(4):465–472, 1923. 13, 14

    1923

  41. [48]

    J. Neyman. On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection.Journal of the Royal Statistical Society, 97(4):558–625, 1934. 13

    1934

  42. [49]

    P. C. O’Brien and T. R. Fleming. A multiple testing procedure for clinical trials.Biometrics, 35(3):549–556, 1979. 4, 6, 17, 20, 24

    1979

  43. [50]

    S. J. Pocock. Group sequential methods in the design and analysis of clinical trials.Biometrika, 64(2):191–199, 1977. 4, 6, 17, 18, 20, 24

    1977

  44. [51]

    Ramdas and R

    A. Ramdas and R. Wang. Hypothesis testing with e-values.Foundations and Trends®in Statistics, 1(1-2):1–390, 2025. 61

    2025

  45. [52]

    H. Robbins. Statistical methods related to the law of the iterated logarithm.The Annals of Mathematical Statistics, 41(5):1397–1409, 1970. 6

    1970

  46. [53]

    Robbins and D

    H. Robbins and D. Siegmund. Boundary crossing probabilities for the Wiener process and sample sums.The Annals of Mathematical Statistics, pages 1410–1429, 1970. 1, 5

    1970

  47. [54]

    D. B. Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology, 66(5):688, 1974. 13, 14

    1974

  48. [55]

    D. O. Scharfstein, A. A. Tsiatis, and J. M. Robins. Semiparametric efficiency and its implication on the design and analysis of group-sequential studies.Journal of the American Statistical Association, 92(440):1342–1350, Dec. 1997. 18

    1997

  49. [56]

    W. F. Stout. The Hartman-Wintner law of the iterated logarithm for martingales.The Annals of Mathematical Statistics, 41(6):2158–2160, 1970. 12

    1970

  50. [57]

    Strassen

    V. Strassen. Almost sure behavior of sums of independent random variables and martingales. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 3,

  51. [58]

    E. O. Taga, S. Oymak, and S. Shekhar. Learning to bet for horizon-aware anytime-valid testing. arXiv preprint arXiv:2603.19551, 2026. 4

    2026 arXiv

  52. [59]

    A. A. Tsiatis, H. Boucher, and K. Kim. Sequential methods for parametric survival models. Biometrika, 82(1):165–173, 1995. 18 27

    1995

  53. [60]

    A. W. Van Der Vaart, J. H. Van Zanten, and others. Reproducing kernel Hilbert spaces of Gaussian priors.IMS Collections, 3:200–222, 2008. 30

    2008

  54. [61]

    Vor´ acek and F

    V. Vor´ acek and F. Orabona. STaR-Bets: Sequential target-recalculating bets for tighter confidence intervals.Advances in Neural Information Processing Systems, 38:161909–161953,

  55. [62]

    A. Wald. Sequential Tests of Statistical Hypotheses.The Annals of Mathematical Statistics, 16 (2):117 – 186, 1945. 61

    1945

  56. [63]

    S. K. Wang and A. A. Tsiatis. Approximately optimal one-parameter boundaries for group sequential trials.Biometrics, pages 193–199, 1987. 17, 18, 19, 20, 21

    1987

  57. [64]

    Waudby-Smith and A

    I. Waudby-Smith and A. Ramdas. Estimating means of bounded random variables by betting. Journal of the Royal Statistical Society Series B: Statistical Methodology, 86(1):1–27, 02 2024. 4

    2024

  58. [65]

    Waudby-Smith, E

    I. Waudby-Smith, E. H. Kennedy, and A. Ramdas. Distribution-uniform anytime-valid sequential inference and the robbins-siegmund distributions.arXiv preprint arXiv:2311.03343, 2023. 1, 2, 3, 4, 5, 10, 29, 31, 37, 43

    2023

  59. [66]

    Waudby-Smith, D

    I. Waudby-Smith, D. Arbour, R. Sinha, E. H. Kennedy, and A. Ramdas. Time-uniform central limit theory and asymptotic confidence sequences.The Annals of Statistics, 52(6):2613–2640,

  60. [67]

    Waudby-Smith, M

    I. Waudby-Smith, M. Larsson, and A. Ramdas. Distribution-uniform strong laws of large numbers.arXiv preprint arXiv:2402.00713, 2024. 9

    2024 arXiv

  61. [69]

    1and supT“ ∆. Then @εą 0, there exists anη

    I. Waudby-Smith, R. Sandoval, and M. I. Jordan. Universal log-optimality for general classes of e-processes and sequential hypothesis tests.arXiv preprint arXiv:2504.02818, 2025. 61 28 A Proofs of the Main Results We use the stochastic ¯onotation suggested in Waudby-Smith et a...

    2025 arXiv

  62. [71]

    ´c 2 and the “canonical

    If Xp0q„Np0,1q, then P ˜ sup sPr1,∆s |Wpsq|s ´1{2ěx ¸ “ ż x ´8 Pz ˜ sup tPr0,log ∆{2s |Xptq|ěx ¸ ϕpzqdz`Φp´xq, and P ˜ sup sPr1,∆s Wpsqs´1{2ěx ¸ “ ż x ´8 Pz ˜ sup tPr0,log ∆{2s Xptqěx ¸ ϕpzqdz`Φp´xq. Therefore, finding distribution functions for q“ 1{2 amounts to finding them ...

  63. [72]

    First, recall a property ofA µpyq

    This leaves us with G0,µpx, yq“B µpyqAµpxq´B µpxqAµpyq, where Bµpyq“ y? 2 M ˆ 1´µ 2 ; 1.5; y2 2 ˙ , and Aµpyq“M ˆ´µ 2 ; 0.5; y2 2 ˙ , whereMis the hypergeometric (Kummer’s) function. First, recall a property ofA µpyq. Lemma B.7.LetA µpyq“Mp ´µ 2 ; 0.5; y2 2q. d dy Aµpyq“´µyM ˆ...

  64. [73]

    power-one

    Therefore, the correlation of the two random variables is´c 1{ a 1`c 2 1 and we can write PpU´c 1Xďc 2, Xăaq´PpU´c 1Xďc 2, Xă´aq“P ´ rUďc 2, Xăa ¯ ´P ´ rUďc 2, Xă´a ¯ “Φ 2 ˜ c2a 1`c 2 1 , a; ´c1a 1`c 2 1 ¸ ´Φ 2 ˜ c2a 1`c 2 1 ,´a; ´c1a 1`c 2 1 ¸ Lemma B.10.LetD νp¨qbe the parab...

  65. [2024]

    1, 2, 3, 4, 5, 6, 9, 22, 36

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.