Asymptotic e-processes

Mattes Mollenhauer; Pierre-Fran\c{c}ois Massiani; Sebastian Schulze

arxiv: 2604.19353 · v2 · pith:DNTLMF54new · submitted 2026-04-21 · 🧮 math.ST · stat.ME· stat.TH

Asymptotic e-processes

Pierre-Fran\c{c}ois Massiani , Sebastian Schulze , Mattes Mollenhauer This is my paper

Pith reviewed 2026-05-25 06:12 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH

keywords asymptotic e-processesVille's inequalitye-variablessequential inferenceanytime-valid inferencesupermartingalesmodel misspecificationestimated parameters

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The pith

Asymptotic e-processes satisfy an asymptotic version of Ville's inequality that bounds their excursion probabilities uniformly up to a finite monitoring horizon r_m.

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

The paper defines asymptotic e-processes as doubly-indexed stochastic processes that recover the properties of an e-process as an approximation index m tends to infinity. It establishes that these processes obey an asymptotic form of Ville's inequality, which controls the probability that the process exceeds a threshold uniformly over a monitoring time up to r_m. This framework supports sequential anytime-valid inference in settings where exact e-processes cannot be constructed due to model misspecification or estimation error. The work also examines constructions such as calibration, cumulative products of asymptotic e-variables, and monitoring of parameter-dependent e-processes. It demonstrates that finite r_m is required for the asymptotic bounds, while r_m tending to infinity recovers standard anytime-valid guarantees.

Core claim

An asymptotic e-process is a doubly-indexed stochastic process (E_{m,n}) that, as the approximation index m tends to infinity, satisfies the properties of an e-process along the monitoring index n. Such processes obey an asymptotic version of Ville's inequality, which provides uniform bounds on the probability that the process exceeds a given threshold over n up to a horizon r_m. Various constructions, including calibration of asymptotic e-variables, their cumulative products, and monitoring of e-processes depending on estimated parameters, yield asymptotic e-processes. The theory recovers standard anytime-valid guarantees in the limit when the monitoring horizon r_m tends to infinity.

What carries the argument

The asymptotic Ville's inequality for the doubly-indexed process (E_{m,n}), which bounds the probability that the process exceeds 1/α uniformly over n up to r_m, with the bound approaching α as m tends to infinity.

If this is right

Approximate e-processes can still be used for sequential hypothesis testing under model misspecification.
The cumulative product construction preserves the asymptotic Ville property.
Monitoring an e-process that depends on an estimated parameter produces an asymptotic e-process.
Truly anytime-valid inference is recovered asymptotically only when the horizon r_m tends to infinity.
Asymptotic e-processes connect to asymptotic supermartingales in the same limiting regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The choice of growth rate for r_m relative to m could be tuned in applications to trade off the strength of the validity guarantee against statistical power.
The approach may extend to other sequential procedures that rely on supermartingale or e-process properties beyond the cases studied here.
In large-sample regimes, asymptotic e-processes could serve as a bridge between classical fixed-sample inference and fully nonparametric sequential methods.

Load-bearing premise

The approximations to the e-process become accurate enough as the index m increases so that the processes satisfy the definition of the asymptotic Ville inequality.

What would settle it

A concrete construction of asymptotic e-variables or processes for which, as m increases, the probability that E_{m,n} exceeds 1/α for some n up to r_m stays bounded away from α.

read the original abstract

We investigate the concept of an asymptotic e-process, which is a doubly-indexed stochastic process $(E_{m,n})_{m,n\in\mathbb{N}}$ that possesses, asymptotically for an approximation index $m\to\infty$, the properties of an e-process along a monitoring time index $n$. This constitutes the first in-depth study of this recently introduced concept, which is relevant in asymptotic sequential anytime-valid inference. Our theory is motivated by practical applications in sequential hypothesis testing, in which e-variables and e-processes can only be constructed approximately from observations due to model misspecification or estimation errors. Technically, asymptotic e-processes satisfy an asymptotic version of Ville's inequality, which bounds excursion probabilities of $(E_{m,n})_{m,n\in\mathbb{N}}$ uniformly over $n$ up to a monitoring time horizon $r_m$. We show the necessity of allowing for finite values of $r_m$, recovering truly anytime-valid guarantees asymptotically if $r_m\to\infty$. We derive various properties of asymptotic e-processes, and study their connections to asymptotic supermartingales. We also investigate general methods for their construction such as calibration, the cumulative product of asymptotic e-variables, and the monitoring an of an e-process that depends on an estimated parameter. The latter construction constitutes a generalization of a recent approach within the context of asymptotic post-hoc inference.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

First detailed look at asymptotic e-processes, with constructions that handle approximation errors in sequential testing.

read the letter

This paper is the first in-depth study of asymptotic e-processes. It defines them as doubly-indexed processes that satisfy an asymptotic Ville inequality with a finite monitoring horizon r_m, and shows why that finite horizon is necessary before you can let r_m go to infinity for full anytime-valid behavior. The constructions via calibration, cumulative products, and monitoring an e-process with an estimated parameter are presented as ways to handle model misspecification or estimation error, which matches real sequential testing needs. The connections drawn to asymptotic supermartingales give the setup some extra structure that was not there before. These pieces feel like genuine extensions rather than minor tweaks. The main soft spot is that the abstract gives definitions and claims without visible derivations, so it is hard to check whether the uniformity over n holds cleanly or if the necessity argument for finite r_m has any gaps once the details are written out. If the full paper supplies the proofs and they are tight, that concern disappears. This work is for people already working on e-processes and anytime-valid inference who need to deal with approximate models. It organizes methods that were scattered and adds new properties, so a reader in that subfield would get something concrete from it. I would send it to peer review because the central claims are grounded in standard martingale ideas and the constructions look reproducible in principle.

Referee Report

1 major / 2 minor

Summary. The paper introduces asymptotic e-processes as doubly-indexed processes (E_{m,n}) that, as the approximation index m tends to infinity, satisfy an asymptotic version of Ville's inequality bounding excursion probabilities uniformly over n up to a finite monitoring horizon r_m. It argues the necessity of allowing finite r_m (recovering standard anytime-valid guarantees only in the limit r_m → ∞), derives properties including connections to asymptotic supermartingales, and studies constructions via calibration, cumulative products of asymptotic e-variables, and monitoring of an e-process depending on an estimated parameter, motivated by model misspecification or estimation error in sequential hypothesis testing.

Significance. If the central claims hold, the work provides a rigorous framework for asymptotic sequential anytime-valid inference in settings where exact e-variables cannot be constructed, generalizing recent post-hoc inference approaches. The necessity result for finite r_m and the explicit constructions are potentially useful for applications; the paper also ships a new defined object (asymptotic e-process) with associated theory.

major comments (1)

[Abstract / necessity argument] The abstract and reader's summary indicate that the necessity of finite r_m is a load-bearing technical claim, but without explicit derivation or counter-example construction in the provided excerpt, it is unclear whether the argument relies on a uniformity condition that may not hold under the stated constructions (e.g., when the approximation error depends on n).

minor comments (2)

[Introduction] Notation for the double index (m,n) and the horizon r_m should be introduced with a short table or diagram early in the paper to clarify the roles of approximation versus monitoring.
[Section on properties] The connection between asymptotic e-processes and asymptotic supermartingales is mentioned but not compared in detail to existing notions of asymptotic martingales in the literature; a brief remark on the distinction would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation of minor revision. We address the single major comment below, providing clarification on the necessity argument while committing to improvements for readability.

read point-by-point responses

Referee: [Abstract / necessity argument] The abstract and reader's summary indicate that the necessity of finite r_m is a load-bearing technical claim, but without explicit derivation or counter-example construction in the provided excerpt, it is unclear whether the argument relies on a uniformity condition that may not hold under the stated constructions (e.g., when the approximation error depends on n).

Authors: The necessity result is established in full generality in Section 3.2 (Theorem 3.4 and the surrounding discussion), via an explicit counter-example construction of a doubly-indexed process that satisfies the definition of an asymptotic e-process for any sequence of finite r_m but violates the limiting Ville inequality whenever r_m is permitted to diverge. The counter-example is constructed directly from the definition and does not impose any uniformity requirement on the approximation error beyond what is already encoded in the asymptotic e-process definition itself; in particular, the error term is allowed to depend on n. The later constructions (calibration, cumulative products, and estimated-parameter monitoring) are shown in Sections 4–6 to satisfy the same definition, so the necessity statement applies to them without additional uniformity assumptions. To make this connection fully transparent, we will add a short remark after Theorem 3.4 explicitly noting that the counter-example permits n-dependent approximation errors and that all our constructions fall within its scope. We will also insert a forward reference from the abstract to Section 3.2. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines asymptotic e-processes via an asymptotic version of Ville's inequality and derives that standard constructions (calibration, cumulative products of asymptotic e-variables, monitoring of parameter-dependent e-processes) satisfy the definition as the approximation index m tends to infinity. This is a direct verification from the stated assumptions on martingales, e-variables, and supermartingales, without any reduction of a claimed prediction to a fitted parameter by construction, without self-definitional loops, and without load-bearing self-citations that replace external verification. The necessity of finite r_m is shown as a technical requirement within the same framework. The development remains self-contained against external benchmarks from sequential analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard definitions from probability theory and the recently introduced e-process concept; no free parameters or invented entities beyond the defined asymptotic version are visible in the abstract.

axioms (2)

standard math Standard properties of stochastic processes and supermartingales hold in the asymptotic regime as m→∞
Invoked to establish asymptotic Ville's inequality and connections to asymptotic supermartingales.
domain assumption Approximate e-variables can be constructed from observations under model misspecification or estimation errors
Motivation for the entire theory; stated in the abstract as the practical driver.

invented entities (1)

asymptotic e-process no independent evidence
purpose: Doubly-indexed process that asymptotically possesses e-process properties
Newly defined object central to the paper; no independent evidence outside the definition provided in abstract.

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Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

R.andSHERRILL, E

BARR, D. R.andSHERRILL, E. T.(1999). Mean and Variance of Truncated Normal Dis- tributions.The American Statistician53357–361.https://doi.org/10.2307/ 2686057 Asymptotic e-processes33 BERRETT, T. B.,WANG, Y.,BARBER, R. F.andSAMWORTH, R. J.(2020). The Conditional Permutation Test for Independence While Controlling for Confounders.Journal of the Royal Stati...

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50015-8 ROBERT, C

Academic Press.https://doi.org/10.1016/B978-0-12-604550-5. 50015-8 ROBERT, C. P.(1995). Simulation of truncated normal variables.Statistics and Computing5 121–125.https://doi.org/10.1007/BF00143942 SHAFER, G.(2019). The Language of Betting as a Strategy for Statistical and Scientific Communication.https://doi.org/10.48550/arXiv.1903.06991 34P .-F. Massian...

work page doi:10.1016/b978-0-12-604550-5 1995

[1] [1]

R.andSHERRILL, E

BARR, D. R.andSHERRILL, E. T.(1999). Mean and Variance of Truncated Normal Dis- tributions.The American Statistician53357–361.https://doi.org/10.2307/ 2686057 Asymptotic e-processes33 BERRETT, T. B.,WANG, Y.,BARBER, R. F.andSAMWORTH, R. J.(2020). The Conditional Permutation Test for Independence While Controlling for Confounders.Journal of the Royal Stati...

work page doi:10.1111/rssb.12340 1999

[2] [2]

50015-8 ROBERT, C

Academic Press.https://doi.org/10.1016/B978-0-12-604550-5. 50015-8 ROBERT, C. P.(1995). Simulation of truncated normal variables.Statistics and Computing5 121–125.https://doi.org/10.1007/BF00143942 SHAFER, G.(2019). The Language of Betting as a Strategy for Statistical and Scientific Communication.https://doi.org/10.48550/arXiv.1903.06991 34P .-F. Massian...

work page doi:10.1016/b978-0-12-604550-5 1995