Asymptotic e-processes

1. Problem Statement

Sequential hypothesis testing relies on e-processes (and their building blocks, e-variables) to provide "Safe Anytime-Valid Inference" (SAVI). Unlike classical p-values, e-processes allow for flexible stopping rules (optional stopping) while maintaining strict Type I error control. Specifically, if the null hypothesis holds, an e-process remains small with high probability; if the null is false, it grows.

However, in practical applications, constructing exact e-processes is often impossible because it requires precise knowledge of unknown quantities (e.g., true distributions or parameters) which must be estimated from data. This leads to model misspecification or estimation errors.

• The Gap: While asymptotic e-variables (sequences of variables converging to e-variables) have been defined, a rigorous framework for asymptotic e-processes that maintains anytime-validity guarantees in the presence of approximation errors has been missing.
• The Naive Approach Failure: A direct generalization of e-processes to the asymptotic setting (requiring Ville's inequality to hold for all time horizons $n$ as the approximation index $m \to \infty$) is too restrictive. Specifically, if an e-variable has a slight excess expectation ($1+\delta_m$), the cumulative product construction (standard for e-processes) causes this error to compound exponentially over time, causing the process to diverge and violating the e-process condition for any fixed finite $m$.

2. Methodology and Core Concept

The authors introduce $r$-asymptotic e-processes, a doubly indexed stochastic process $(E_{m,n})_{m,n \in \mathbb{N}}$, where:

• $n$: The monitoring time (sequential observation index).
• $m$: The approximation index (representing the quality of the estimator or model, where $m \to \infty$ implies better approximation).

Key Innovation: The Monitoring Horizon $r_m$
Instead of demanding validity for all time $n \in \mathbb{N}$, the authors define the process as valid only up to a time horizon $r_m$ that depends on the approximation quality $m$.

• Intuition: As the approximation improves ($m \to \infty$), the allowed monitoring horizon $r_m$ expands ($r_m \to \infty$).
• Mechanism: The horizon $r_m$ is chosen such that the accumulated approximation error over the interval $[0, r_m]$ remains negligible. This prevents the "compounding error" problem inherent in cumulative products.

Formal Definition:
A process $E$ is an $r$-asymptotic e-process if, for any sequence of stopping times $\tau = (\tau_m)$ bounded by $r_m$, the sequence $(E_{m, \tau_m})_m$ is a uniformly strongly asymptotic e-variable (i.e., $\limsup_{m \to \infty} \sup_{P \in \mathcal{P}} \mathbb{E}_P[E_{m, \tau_m}] \le 1$).

3. Key Contributions and Theoretical Results

A. Characterization and Equivalence

• Theorem 3.3: Shows that if a process is an $r$-asymptotic e-process for any bounded sequence $r$, it is also an $r$-asymptotic e-process for a sequence where $r_m \to \infty$. This justifies focusing on diverging horizons.
• Theorem 3.4 (Supermartingale Connection): Establishes that $r$-asymptotic e-processes are equivalent to processes that are upper-bounded by a supermartingale (up to time $r_m$) that satisfies an asymptotic calibration condition. This generalizes the classical result that e-processes are bounded by test supermartingales.
• Theorem 3.7 (Limit Behavior): Proves that if a sequence of processes converges in $L^1$ to a limit process $F$, then the sequence is an $r$-asymptotic e-process (with $r_m \to \infty$) if and only if the limit $F$ is a true e-process.

B. Asymptotic Supermartingales (ASMs)

The authors define Asymptotic Supermartingales (ASMs) and the Asymptotic Supermartingale Property (ASP).

• ASP: A weaker condition than $L^1$ convergence to a supermartingale. It requires that the "drift" (excess expectation) vanishes asymptotically: $\lim_{m \to \infty} \sup_{P} \mathbb{E}_P[(\mathbb{E}[E_{m,n+1}|\mathcal{F}_{m,n}] - E_{m,n})^+] = 0$.
• Theorem 4.6: Provides a sufficient condition for a process to be an $r$-asymptotic e-process. If a process has the ASP and is asymptotically calibrated, it is an $r$-asymptotic e-process for any sequence $r$ satisfying:
  $$\lim_{m \to \infty} \sup_{P} \mathbb{E}_P \left[ \sum_{n=0}^{r_m-1} \delta^+_{m,n} \right] = 0$$
  This formalizes the intuition that $r_m$ must grow slowly enough to prevent error accumulation.

C. Asymptotic Ville's Inequality

Theorem 5.1 is the central result for practical application. It states that for an $r$-asymptotic e-process $E$ and any threshold $1/\alpha$:
$$\limsup_{m \to \infty} \sup_{P \in \mathcal{P}} P \left( \sup_{n \in \{1, \dots, r_m\}} E_{m,n} \ge \frac{1}{\alpha} \right) \le \alpha$$
This guarantees that the probability of the process crossing the threshold (Type I error) is bounded by $\alpha$ asymptotically, provided the monitoring stops before $r_m$.

D. Construction Methods

The paper demonstrates how to construct these processes from asymptotic e-variables:

1. Cumulative Product: Standard construction $E_{m,n} = \prod_{i=0}^n e_{m,i}$. The authors show this works if the approximation error $\delta_m$ decays fast enough relative to $r_m$ (specifically $r_m \cdot \delta_m \to 0$).
2. Time Mixture: A weighted average of e-variables. The paper notes that time-mixture e-processes often do not satisfy the ASP (they are not supermartingales), yet they can still be asymptotic e-processes, highlighting the generality of the new definition.
3. Calibration of Anytime p-values: The authors show how to convert P-uniform anytime p-values (from Waudby-Smith et al., 2026) into asymptotic e-processes. They identify that the standard definition of anytime p-values yields a "uniformly asymptotic" e-variable, which is insufficient for their framework. They propose a strengthened definition (P-uniform strongly anytime p-values) to ensure the resulting e-process is valid.

4. Results and Numerical Validation

• Counterexamples: The authors demonstrate that naive diagonal cumulative products (using $e_{i,i}$) fail because past errors are never forgotten.
• Simulation (Section 7): They simulate a cumulative product process where the e-variables have a mean of $1 + d_m$ ($d_m \to 0$).
  • They set the horizon $r_m = \lfloor 4m^{0.5} \rfloor$.
  • Results show that as $m$ increases, the empirical probability of crossing the threshold $1/\alpha$ before time $r_m$ converges to $\le \alpha$.
  • Trajectories crossing the threshold after $r_m$ are frequent, illustrating the necessity of the finite horizon $r_m$ for finite $m$.

5. Significance and Impact

• Bridging Theory and Practice: This work resolves the tension between the theoretical elegance of e-processes (which require exact models) and the practical reality of estimation errors. It allows practitioners to use approximate models in sequential testing while retaining rigorous error bounds.
• Refining SAVI: It extends the Safe Anytime-Valid Inference framework to asymptotic settings, a crucial step for large-scale data applications where exact distributions are unknown.
• Flexibility: By introducing the horizon $r_m$, the framework accommodates a wider class of processes (including those that are not supermartingales) and provides a tunable trade-off between approximation quality and monitoring duration.
• Foundational for Future Work: The paper sets the stage for investigating Type II error (power) under alternatives and exploring weaker notions of asymptotic e-variables, moving the field toward more robust, real-world sequential analysis tools.

In summary, the paper provides a mathematically rigorous and practically applicable framework for asymptotic e-processes, defining a time horizon $r_m$ that scales with approximation quality to ensure valid sequential hypothesis testing even when models are imperfect.