E-Process in Sequential Inference

• E-processes are nonnegative adapted processes defined such that their value at every finite stopping time is an e-value, ensuring an expected value at most 1.
• They operationalize sequential tests by thresholding at 1/α, linking evidential statistics with Ville’s inequality to guarantee Type-I validity.
• Aggregated WAIT e-processes leverage weighted levels to achieve asymptotically log-optimal growth, guiding the design of powerful sequential tests.

An e-process is a nonnegative adapted process whose value at every finite stopping time is an e-value, that is, a nonnegative random variable with null expectation at most $1$. In discrete-time sequential inference, this optional-stopping formulation makes e-processes a canonical evidential object: thresholding an e-process at $1/\alpha$ yields a level-$\alpha$ sequential test, and recent work shows not only that every valid sequential test can be represented by thresholding some e-process, but also that asymptotically optimal tests can be aggregated into asymptotically log-optimal e-processes through WAIT constructions. A related asymptotic theory extends the concept to bi-indexed approximate processes $E_{m,n}$ that recover e-process behavior only in the limit as an approximation parameter $m\to\infty$ (Ram et al., 12 May 2026, Massiani et al., 21 Apr 2026).

1. Formal definition and probabilistic framework

The basic setting is a discrete-time filtered probability space

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$

with $\mathbb N_0=\{0,1,2,\dots\}$, and a composite null hypothesis represented by a family $\mathcal P$ of probability measures on $(\Omega,\mathcal F)$. A sequential test at level $\alpha$ is a stopping time $1/\alpha$0 such that

$1/\alpha$1

An e-value is a nonnegative random variable $1/\alpha$2 satisfying

$1/\alpha$3

The sequential analogue is an e-process. In the discrete-time definition adopted in the recent optimality literature, a nonnegative adapted process $1/\alpha$4 is a $1/\alpha$5-e-process if

$1/\alpha$6

Equivalently, $1/\alpha$7 is an e-value for every finite stopping time $1/\alpha$8.

Two technical features are emphasized in this formulation. First, an e-process may start at $1/\alpha$9 and need not be strictly positive. Second, strict positivity can be imposed without changing asymptotic log-growth by the transformation

$\alpha$0

since $\alpha$1 for every finite stopping time $\alpha$2, and if $\alpha$3, then the asymptotic quantity $\alpha$4 is unchanged (Ram et al., 12 May 2026).

2. Thresholding, Ville validity, and representation of sequential tests

The operational link between e-processes and testing is thresholding. For a $\alpha$5-e-process $\alpha$6 and $\alpha$7, define

$\alpha$8

Ville’s inequality for nonnegative supermartingales then yields

$\alpha$9

Thus every e-process induces a family of level-$E_{m,n}$0 sequential tests.

A complementary completeness statement goes in the opposite direction: every level-$E_{m,n}$1 sequential test can be recovered by thresholding some e-process at $E_{m,n}$2. At the level of Type-I validity, e-processes therefore form a complete representation of sequential tests. The statistical interpretation is that a sequential test is a stopping rule, whereas an e-process is a running evidence or capital process against the null; rejection occurs when the accumulated e-evidence reaches $E_{m,n}$3.

This representational equivalence does not, by itself, impose any efficiency requirement. A test may be encoded by a crude process—for example, by an indicator-like construction that remains near $E_{m,n}$4 until the stopping time and then jumps—without having useful growth behavior under alternatives. The recent optimality theory is concerned precisely with when such representations can be made asymptotically efficient on the logarithmic scale (Ram et al., 12 May 2026).

3. Optimality notions for tests and e-processes

For a decreasing sequence of levels $E_{m,n}$5, it is standard to set

$E_{m,n}$6

so that $E_{m,n}$7. Under an alternative distribution $E_{m,n}$8 and benchmark information rate $E_{m,n}$9, the strongest test optimality notion used in the recent converse theorem is the almost-sure first-order rate

$m\to\infty$0

The same work distinguishes weaker notions: rate in probability, $m\to\infty$1-rate, and expectation-rate. The implications are asymmetric. Almost-sure rate implies in-probability rate; $m\to\infty$2-rate implies both in-probability rate and expectation optimality; expectation-rate alone is substantially weaker.

For e-processes, the corresponding efficiency criterion is logarithmic growth under an alternative. If $m\to\infty$3, its log-growth rate under $m\to\infty$4 is the limit, when it exists,

$m\to\infty$5

If this limit equals $m\to\infty$6, the process is log-optimal under $m\to\infty$7. More generally, for $m\to\infty$8, an e-process is $m\to\infty$9-log-efficient if the rate equals $$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$0.

Threshold tests inherit these growth properties. If $$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$1 is nondecreasing, finite-valued, and satisfies

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$2

almost surely or in probability under $$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$3, then its threshold times

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$4

satisfy

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$5

with the same mode of convergence. This gives one direction of optimality transfer: log-optimal e-processes yield asymptotically optimal sequential tests. The converse direction is subtler and depends on explicit aggregation constructions.

A major nuance is that expectation optimality alone does not control aggregated growth. An appendix counterexample constructs tests with

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$6

but without convergence in probability of $$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$7; the associated aggregate then has a random limiting log-rate $$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$8 for a nondegenerate random variable $$(\Omega,\mathcal F,(\mathcal F_t)_{t\in\mathbb N_0}),$$9, rather than a deterministic limit (Ram et al., 12 May 2026).

4. WAIT e-processes and the converse optimality theorem

The main constructive device is the WAIT e-process, where WAIT stands for Weighted Aggregates of Indicators of stopping Times. Let $\mathbb N_0=\{0,1,2,\dots\}$0 be level-$\mathbb N_0=\{0,1,2,\dots\}$1 sequential tests satisfying

$\mathbb N_0=\{0,1,2,\dots\}$2

and choose nonnegative weights $\mathbb N_0=\{0,1,2,\dots\}$3, not all zero, subject to the budget constraint

$\mathbb N_0=\{0,1,2,\dots\}$4

The associated aggregate is

$\mathbb N_0=\{0,1,2,\dots\}$5

This process is nondecreasing, starts at $\mathbb N_0=\{0,1,2,\dots\}$6, and is a valid $\mathbb N_0=\{0,1,2,\dots\}$7-e-process. The proof is direct: $\mathbb N_0=\{0,1,2,\dots\}$8 for every finite stopping time $\mathbb N_0=\{0,1,2,\dots\}$9. The deterministic object governing its asymptotic growth is the cumulative weight profile

$\mathcal P$0

Under the budget constraint, one always has $\mathcal P$1, so whenever

$\mathcal P$2

necessarily $\mathcal P$3.

The central theorem states that if the tests have almost-sure rate $\mathcal P$4 under an alternative $\mathcal P$5,

$\mathcal P$6

and the profile satisfies $\mathcal P$7 and

$\mathcal P$8

then the WAIT aggregate obeys

$\mathcal P$9

This yields the converse to the earlier thresholding direction: asymptotically optimal tests can be aggregated into asymptotically log-optimal e-processes. When the profile is full-rate, $(\Omega,\mathcal F)$0, the WAIT process is log-optimal. The proof is based on a pathwise sandwich that compares $(\Omega,\mathcal F)$1 to $(\Omega,\mathcal F)$2 up to finite additive constants (Ram et al., 12 May 2026).

5. Weight schedules, profile design, and explicit rates

The WAIT theorem reduces growth design to the choice of levels $(\Omega,\mathcal F)$3 and weights $(\Omega,\mathcal F)$4. In the unweighted case $(\Omega,\mathcal F)$5, the relevant profile becomes

$(\Omega,\mathcal F)$6

and if

$(\Omega,\mathcal F)$7

then the counting process

$(\Omega,\mathcal F)$8

satisfies $(\Omega,\mathcal F)$9.

Several explicit schedules illustrate how profile design governs efficiency. With dyadic levels $\alpha$0, one has $\alpha$1, hence $\alpha$2; the resulting unweighted aggregate has log-rate $\alpha$3. With power-law levels $\alpha$4, one gets $\alpha$5, so $\alpha$6, giving log-growth rate $\alpha$7. By contrast, log-corrected schedules

$\alpha$8

satisfy $\alpha$9 and $1/\alpha$00; the corresponding unweighted process achieves the full rate $1/\alpha$01. An iterated-log schedule of the form

$1/\alpha$02

also attains full rate.

A notable weighted construction rescues the dyadic grid. For $1/\alpha$03, choosing

$1/\alpha$04

saturates the validity budget, since $1/\alpha$05, and produces a profile with $1/\alpha$06. The resulting weighted dyadic WAIT process is therefore a $1/\alpha$07-e-process with

$1/\alpha$08

showing that geometric level schedules need not be intrinsically suboptimal; the decisive factor is the cumulative weight profile rather than the level grid alone (Ram et al., 12 May 2026).

6. Asymptotic e-processes and approximate sequential validity

An asymptotic e-process is a bi-indexed process

$1/\alpha$09

designed for settings in which exact e-variables are unavailable because of model misspecification, nuisance estimation, or other approximation error. Here $1/\alpha$10 is an approximation index and $1/\alpha$11 is the monitoring time. Because approximation error can accumulate over $1/\alpha$12, validity is indexed by a monitoring horizon sequence $1/\alpha$13.

The defining notion is uniform strong $1/\alpha$14-asymptotic validity. If $1/\alpha$15 ranges over stopping-time sequences with $1/\alpha$16 almost surely under every null $1/\alpha$17, then $1/\alpha$18 is a uniformly strongly $1/\alpha$19-asymptotic e-process when

$1/\alpha$20

This is the asymptotic counterpart of the optional-stopping definition of an exact e-process.

The corresponding asymptotic Ville inequality states that for every $1/\alpha$21,

$1/\alpha$22

Thus one obtains asymptotic anytime-validity, but only up to the horizon $1/\alpha$23 permitted by the approximation quality.

A major structural tool is the asymptotic supermartingale property (ASP), defined through the conditional drift

$1/\alpha$24

If for each fixed $1/\alpha$25,

$1/\alpha$26

and the process is asymptotically calibrated at time $1/\alpha$27, then $1/\alpha$28 is an $1/\alpha$29-asymptotic e-process for any horizon sequence satisfying

$1/\alpha$30

For cumulative-product constructions from approximate e-variables with per-step conditional excess bounded by $1/\alpha$31, this reduces to the explicit requirement

$1/\alpha$32

The asymptotic theory also clarifies scope and limits. If $1/\alpha$33 in $1/\alpha$34 uniformly in $1/\alpha$35 for each fixed $1/\alpha$36, then the existence of some diverging $1/\alpha$37 for which $1/\alpha$38 is $1/\alpha$39-asymptotic is equivalent to the limit process $1/\alpha$40 being an exact e-process. At the same time, not every asymptotic e-process has the ASP, just as not every exact e-process is a supermartingale. In particular, time-mixture constructions can define asymptotic e-processes whose limits are exact e-processes but not supermartingales. This shows that optional-stopping validity and supermartingale structure, while closely related, are not identical notions (Massiani et al., 21 Apr 2026).