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The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent
Patrick Forré
TL;DR
Type-II error is horizon-dependent, and the paper examines both a catastrophic-bet construction and exponent bounds. It relates the exact error ordering of flattened designs to the horizon while distinguishing the general test-supermartingale argument from results requiring independence.
Problem
Type-II error depends on the horizon and on features not optimized by design criteria based on e-power.
Source: The asymmetry · p. 3Method
The paper pairs a catastrophic first-bet construction with a conditional one-step exponent argument for test supermartingales.
Source: §2.5 · p. 9, §3.2 · p. 12Results
For flattened designs, the exact type-II error is nondecreasing in B_t, so minimizing B_t orders their errors exactly.
Source: §5 · p. 27Takeaways & Limitations
The ordering by B_t remains the design criterion even when exp(B_t) does not fall below one.
Source: §5 · p. 27The analysis assumes simple, known null and alternative hypotheses and does not treat composite testing.
Source: Scope · p. 5
Abstract
from arXiv · showhide
In safe hypothesis testing with test supermartingales, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $α\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $1/α$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate $\mathbb{E}_{P_1}[\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $α$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $Λ(E)=\sup_{s\ge0}\{-\log \mathbb{E}_{P_1}[E^{-s}]\}$, whose range is exactly determined: $\sup_E Λ(E)=\mathrm{KL}(P_0\|P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\mathrm{KL}(P_1\|P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^β/\mathbb{E}_{P_0}[R^β]$: the ceiling as $β\downarrow0$, $R$ at $β=1$. Which $β$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\log(1/α)/\mathrm{KL}(P_1\|P_0)$ alone, beaten by sharpening $(β>1)$ below it and by flattening above.
1 Introduction
The paper shows that e-power alone gives no uniform finite-horizon type-II guarantee, then identifies an exponent ceiling and a horizon-dependent choice of design. It distinguishes what holds for arbitrary test supermartingales from what is matched in the i.i.d. case.
The asymmetry
- Type-I error is controlled uniformly over time, whereas type-II error is horizon-dependent and is not controlled by the same framework structure.The paper distinguishes fixed-horizon error from the probability of not having rejected by that horizon.Source: The asymmetry · p. 3
First: e-power alone bounds nothing
- For every positive conditional e-power, level, and finite horizon, some testing problem has probability of non-rejection arbitrarily close to one.The construction uses a catastrophic first bet; a fixed i.i.d. product with positive e-power nevertheless rejects eventually almost surely.Source: First: e-power alone bounds nothing · p. 3
Second: the exponent has an exact ceiling
- The Chernoff–Stein exponent has ceiling KL(P0||P1), approached but not attained, while its upper bound applies stepwise to arbitrary test supermartingales.In the i.i.d. case the bound matches the type-II exponent, and no member attains the ceiling under the stated non-degeneracy conditions.Source: Second: the exponent has an exact… · p. 4
The dichotomy, and what to bet
- The e-power ceiling is KL(P1||P0), attained uniquely up to P0-null modification by the likelihood ratio R; the exponent ceiling is the reverse divergence and is unattained.Along the flattened family, e-power reaches its ceiling at β=1 while the exponent ceiling is approached as β tends to zero.Source: The dichotomy, and what to bet · p. 4
- R is horizon-optimal within the flattened family only at t=log(1/α)/KL(P1||P0); sharpening wins below that horizon and flattening above it.The crossover is the number of observations R needs to accumulate log(1/α) nats of evidence on average.Source: The dichotomy, and what to bet · p. 5
Scope
- The analysis assumes simple, known null and alternative hypotheses; it does not treat the composite case.The designs approaching KL(P0||P1) are built from the likelihood ratio R.Source: Scope · p. 5
2 Setup and notation
The setup defines betting and rejection on filtered spaces, then distinguishes horizon-indexed type-II errors from type-I control. It introduces e-power and the Chernoff–Stein exponent as different summaries, with the latter governing type-II error exponents.
2 Setup and notation
- The paper fixes extended-real conventions for logarithms and negative powers, including their values at zero and infinity.These conventions apply throughout the notation and formulas.Source: §2 · p. 5
2.1 The filtered setting
- The setting uses a filtration without assumptions on how it arises; the i.i.d. model is invoked only for statements that name it.In the i.i.d. model, the standing divergence assumption is imposed at each finite time and corresponds to finite per-observation KL divergence.Source: §2.1 · pp. 5–6
2.2 e-variables and test supermartingales
- Conditional e-bets have null conditional expectation at most one, so their product wealth is a nonnegative supermartingale and threshold crossing has type-I error at most α over all times.The rejection threshold is 1/α.Source: §2.2 · p. 7
2.3 The type-II functionals
- Type-II error is indexed by a finite horizon; for the i.i.d. products considered, its value at an infinite horizon is zero when e-power is positive.For general betting sequences, the probability of never rejecting need not vanish, so it does not provide a horizon-free statement.Source: §2.3 · p. 8
2.4 The two scalars attached to an e-variable
- E-power, the mean of log E under P1, is bounded by KL(P1||P0) and reaches equality only when E equals the likelihood ratio R, up to P0-null modification.The paper contrasts this mean with Λ(E), which concerns the lower tail of log E.Source: §2.4 · p. 9
2.5 Why Λ is the exponent
- The optimized tilted exponent Λ(E) is the type-II exponent for an i.i.d. product under the stated conditions, while a conditional Chernoff bound applies to every test supermartingale.The matching result requires positive e-power and the specified rate-function regularity conditions.Source: §2.5 · pp. 9–10
2.6 Notation index
- The notation index maps symbols for divergences, flattened likelihood ratios, and horizon-optimal tilts to their definitions and uses.It also distinguishes the tilted exponent of an e-variable from the optimized t-step exponent of a test supermartingale.Source: §2.6 · p. 11
3 e-power alone does not bound the type-II error
Positive e-power drives eventual rejection in the i.i.d. setting, but by itself gives no uniform finite-horizon type-II bound. A construction explains why, while lower-tail control supplies quantitative bounds.
3.1 The heuristic, and the exact point at which it fails
- Expected log-wealth grows at least linearly under positive e-power, but a large lower tail can still prevent threshold crossing with high probability.The expectation can be driven by very large gains on a low-probability event while wealth remains too low on its complement.Source: §3.1 · pp. 11–12
3.2 The construction
- A deep first-step crash can keep the wealth below threshold through t while every bet remains a conditional e-variable and has conditional e-power c.On the crash event the partial maximum stays at or below the threshold; each step's conditional e-power is exactly c.Source: §3.2 · pp. 12–13
- For every c>0, α∈(0,1), and finite t, conditional e-power exactly c can coexist with P1(τα>t) arbitrarily close to one, precluding a uniform bound below one.The construction varies the testing problem as well as the design, so the result concerns uniformity over models.Source: §3.2 · pp. 13–14
3.3 A worked instance
- At t=10, the worked example has power exactly 1/2 despite expected log-wealth of 1.0 nats and e-power 0.1 per step.It rejects at t=1 when X1=1 and does not reject by t=10 when X1=0.Source: §3.3 · p. 14
3.4 What the theorem does not say
- In the i.i.d. setting, positive e-power ensures eventual rejection almost surely, but this pointwise asymptotic result does not give uniform finite-horizon control.The theorem fixes a design and lets t grow; the counterexample fixes t and varies the design.Source: §3.4 · p. 15
- A lower-tail bound can restore an exponential type-II bound, and in the i.i.d. case positive e-power plus Cramér’s moment condition characterizes exponential decay.With a conditional inverse moment bound, the error is at most exp(s0ℓ−tc0); finite variance alone can instead yield an O(1/t) bound.Source: §3.4 · pp. 15–16
4 The ceiling: matching upper and lower bounds
This section develops the upper ceiling for the Chernoff–Stein exponent, using conditional Hölder bounds for test supermartingales on arbitrary filtered spaces and identifying the i.i.d. ceiling as KL(P0||P1). For i.i.d. testing with finite KL(P0||P1), it also shows how the ceiling constrains actual type-II error through level-α acceptance regions and a limit of the flattened likelihood-ratio family.
4.1 The Rényi quantities
- In the i.i.d. model, λ(β)=D_{1−β}(P0||P1) for 0<β<1; λ decreases strictly unless log R is P0-a.s. constant, and λ(1)=0.Source: §4.1 · p. 17
4.2 The conditional Rényi quantities
- Conditionally on the past, λ_t(β) is a nonnegative, nonincreasing stepwise profile; in the i.i.d. model it equals the constant λ(β) P1-a.s.Source: §4.2 · pp. 17–18
4.3 The upper bound
- Conditional Hölder, applied once per step, bounds every test-supermartingale exponent by the average per-step conditional Chernoff–Stein ceilings; in the i.i.d. case this is KL(P0||P1).The bound applies without product structure in the general filtered setting.Source: §4.3 · pp. 18, 21
- The general cap is tight per step, but matching a deterministic total requires nonrandom per-step values; without that condition, sharpness remains open, and the bound is on the exponent, not directly the error.Source: §4.3 · pp. 21–22
- In the i.i.d. model with finite KL(P0||P1), the Chernoff–Stein ceiling constrains actual type-II error for every betting sequence and α.The argument treats the betting sequence’s acceptance region as a level-α test and compares its error with the least type-II error achievable at horizon t [Che56], [CT06, Thm. 11.8.3],.Source: §4.3 · p. 22
- As β↓0, the flattened family’s h_β converges pointwise to log R, with an integrable majorant R−1 enabling passage to the limiting expectation.The limit is E_P0[log R] = −KL(P0||P1) when P0≪P1; both sides are +∞ otherwise.Source: §4.3 · p. 22
4.4 The lower bound
Flattened likelihood-ratio designs provide lower-bound witnesses for the exponent ceiling, whose supremum is KL(P0‖P1) in the i.i.d. setting. The general filtered-space results have a conditional, per-step form rather than one fixed scalar ceiling.
4.4 The lower bound
- The i.i.d. exponent ceiling is matched but unattained, whereas the e-power ceiling KL(P1‖P0) is attained by its optimal design.These ceilings are not comparable to one another.Source: §4.4 · p. 23
- As β↓0, the flattened-design exponent bound tends to KL(P0‖P1), giving the lower bound on the supremum of Λ(E) over e-variables.For each β∈(0,1], Rβ is an e-variable attaining its corresponding bound.Source: §4.4 · pp. 23–24
- The general filtered-space ceiling is an average of per-step conditional Chernoff–Stein exponents, rather than one fixed number.Aggregation uses an essential supremum; the i.i.d. setting removes that complication and yields the scalar KL(P0‖P1) ceiling.Source: §4.4 · p. 24
4.5 Non-attainment
Under the stated assumptions, the optimized exponent has a finite maximizing tilt when the wealth is nonconstant, but the ceiling is not attained at any finite horizon. The ceiling is nevertheless matched by i.i.d. products, and the two divergence-based objectives have different optimizers.
4.5 Non-attainment
- With P0 ≪ P1 and P0(Wt ≠ 1) > 0, the optimized t-step exponent attains its supremum at a finite tilt.The ceiling itself remains unattained under the non-degeneracy assumptions stated for Proposition 4.24.Source: §4.5 · pp. 24–25, §4.6 · p. 25
4.6 The two-sided statement
- For every finite horizon, the supremum of the optimized exponent over test supermartingales equals KL(P0‖P1), is matched by i.i.d. products, and is attained by no design under the stated conditions.The common value is approached along the flattened family as β↓0.Source: §4.6 · pp. 25–26
4.7 The two ceilings
- The e-power ceiling is KL(P1‖P0), attained uniquely up to P0-null modification by the likelihood ratio R; the exponent ceiling is KL(P0‖P1) and is unattained under the stated conditions.The two optima evaluate the divergence in opposite argument orders.Source: §4.7 · p. 26
- Neither KL(P0‖P1) nor KL(P1‖P0) necessarily dominates the other; their relative size depends on the model.For the Bernoulli example, the divergences are 1.6145 and 0.6371, respectively; exchanging the hypotheses reverses the comparison.Source: §4.7 · p. 26
4.8 A worked instance
- In the worked instance, β=0.1 and β=0.05 yield exponents within 10% and 5% of the ceiling, respectively, but no e-variable reaches 0.0204110.The final column is a limit, not an e-variable.Source: §4.8 · p. 27
- The maximal-e-power likelihood ratio has e-power 0.0201355 but realizes barely a quarter of the largest possible exponent, and the flattening R^β* already beats it.The example’s optimized tilt is β*=0.6659.Source: §4.8 · p. 27
5 What to bet at a finite horizon
This section examines finite-horizon betting choices, including how the best tilt in the flattened likelihood-ratio family depends on the horizon and a predictable, wealth-adaptive tilt in the reported Gaussian example. It also describes the adaptive rule as a heuristic specific to a fixed horizon and target, improving the error constant rather than the certified exponent.
5.1 The two scalars along the family
- Along the flattened family, e-power rises to KL(P1||P0) at β = 1, while λ(β) falls from its β↓0 ceiling KL(P0||P1) to 0.The e-power falls to 0 as β↓0, so no family member is optimal for both objectives.Source: §5.1 · p. 28
5.2 The flattened family consists of likelihood-ratio tests
- Every flattened design tests the same log-likelihood-ratio sum at a β-dependent threshold, and its exact type-II error is nondecreasing in that threshold.For β > 1, the surrogate bound is vacuous, but the threshold still orders exact errors and the optimum can exceed 1 below the crossover.Source: §5 · p. 27, §5.2 · p. 29
5.3 The horizon-optimal tilt
- The horizon-optimal tilt is the likelihood ratio at t = log(1/α)/KL(P1||P0), is sharper below that horizon, and flatter above it.As t increases, the optimal β decreases toward 0.Source: §5.3 · p. 30
5.4 The price of the ceiling
- Buying an exponent within ε of the ceiling costs a tilt of order ε and a horizon of order ε^-1 before the test gains power.For R^β, power is at most O(β) through horizons t ≤ ℓ/(2cβ) when cβℓ > 2vβ.Source: §5.4 · p. 32
5.5 The Gaussian model in closed form
- In the Gaussian model, λ(β) = (1 − β)D and cβ = β(2 − β)D, while β*(t) = ℓ/(tD) exactly and β = 1 achieves one-quarter of the ceiling.The growth-optimal design is beaten after the crossover t = ℓ/D, with the horizon-optimal exponent tending to tD.Source: §5.5 · p. 33
5.6 Adapting the tilt to the wealth
- A predictable rolling-horizon tilt re-solves the residual threshold problem at each step and bets harder when behind or time is short, then coasts when ahead.The tilt is measurable with respect to the prior information, so the resulting sequence is a betting sequence.Source: §5.6 · pp. 33–34
- Equation (32) can be solved by bisection because g is increasing, with safeguarded Newton steps converging quickly; when κ lacks a closed form, its expectations use the known null P0.Quadrature or Monte Carlo computes these expectations, and a single P0 sample drawn once can be reused at every β and step.Source: §5.6 · p. 34
- In the Gaussian Δ=1, α=0.05 grid, the rolling-horizon tilt captures 95.6%, 97.6%, and 99.1% of the attainable gain over the best constant β.The rule minimizes a bound on residual error, not the error itself; it is not exactly optimal, though it numerically recovers almost all the finite-horizon control optimum achieves.Source: §5.6 · p. 34
- Allowing β > 1 improves the optimal rule’s error at T=20 from 6.46 · 10^-3 to 5.93 · 10^-3; capping the rolling tilt at 1 multiplies error by 1.10 to 1.11 across T=5, 10, 20, and 50.The adaptive rule chooses β > 1 when the residual gap per remaining bet exceeds the largest attainable e-power; raising β trades lower drift for greater dispersion when the design is behind.Source: §5.6 · p. 35
- The rule is horizon-specific, requires numerical error evaluation, and cannot raise the certified exponent above its ceiling; its gain is in the constant, not the rate.It assumes a fixed horizon and target γ_T; for γ̄_T the residual problem has an absorbing region above the threshold, so (31) is no longer the right surrogate.Source: §5.6 · p. 35
6 Discussion
The discussion distinguishes the e-power used in sample-size heuristics from the quantity governing type-II error, and identifies limits on the results’ scope. It also relates the design trade-off to sequential analysis.
6 Discussion
- The Chernoff–Stein ceiling is an oracle quantity for simple hypotheses; its approaching designs use the true likelihood ratio, so the composite-alternative analogue remains open.The paper does not determine how much of inf_{P1 in H1} KL(P0||P1) a design can access without knowing which alternative obtains.Source: §6 · p. 35
- Unlike the SPRT, a test supermartingale fixes the upper threshold at 1/α to retain type-I validity at every stopping time, constraining the available design trade-off.The paper identifies the Chernoff–Stein ceiling as the price of this constraint and describes what remains free through horizon-dependent design.Source: §6 · p. 35
- The finite-horizon results do not determine exact second-order behavior, including the O(1) constant, Bahadur–Rao prefactor, or a matching converse; misspecification is also untreated.The upper bounds remain valid for arbitrary e-variables, but attainment relies on designs built from the true likelihood ratio.Source: §6 · p. 36
- The e-power sample-size heuristic locates the horizon where a threshold crosses the mean of the log-likelihood ratio, but e-power alone gives no uniform finite-horizon guarantee.A lower-tail condition, rather than the mean growth rate alone, is needed for explicit finite-horizon bounds.Source: §6 · p. 36
- In general filtered settings, the ceiling need not be sharp: per-step bounds may fail to aggregate, and the sequential-error exponent is known only through a lim inf.The non-random conditional-divergence assumption is sufficient, not characterized; whether the essential suprema can be strict is open.Source: §6 · p. 36
- Finite-horizon certification and composite hypotheses are left to companion work; the paper establishes what the exponent can be, not what a given design can certify.The companion paper treats lower-tail conditions under arbitrary filtrations without product structure.Source: §6 · p. 36
A Beyond the i.i.d. model
This appendix collects replacements for results proved under the i.i.d. model, together with the hypotheses each replacement requires. Table 1 serves as their index.
A Beyond the i.i.d. model
- The appendix states which i.i.d.-model results have general-setting replacements and the hypotheses those replacements require.It explicitly does not assume the i.i.d. model; Table 1 indexes the results and their costs.Source: A Beyond the i.i.d. model · p. 38, Table 1 · p. 38
- The appendix leaves Theorem 5.6, the strictness clause of Corollary 4.18, and Proposition 5.5(b)–(c) without general forms.Lemma 4.22 is reused under its stated conditions: equivalence of the restricted measures and positive probability that W_t differs from 1 under P0.Source: A Beyond the i.i.d. model · p. 38
A.1 Free of any extra hypothesis
This subsection gives conditional results that need no extra hypothesis, including a flattened design attaining a one-step bound and an equality characterization. It also states a support condition for identifying a one-step Chernoff quantity.
A.1 Free of any extra hypothesis
- Equality in the conditional bound forces the e-variable to equal the one-step likelihood ratio P1-a.s.The proof passage states that equality forces E_n/R_n=1; the conditional relative entropy is KL_n.Source: §A.1 · p. 39
- For β in (0,1], the conditional flattened design is a conditional e-variable and attains the one-step bound; equality holds exactly for that design.Across steps through time t, equality requires E_n=R_{β,n} P1-a.s. at every step.Source: §A.1 · p. 39
- The one-step Chernoff information identity assumes P0(R_n>0|F_{n−1})=1; without it, the left side can be infinite while the right side is finite.The supplied Bernoulli example gives left side +∞ and right side log 2.Source: §A.1 · p. 40
- The conditional Chernoff quantity can be expressed through the flattened normalizer Z_{β,n}, with the supremum reduced to β in (0,1].The passage derives this using convexity of log Z_{β,n} and its values at β=0 and β=1.Source: §A.1 · p. 40
A.2 Deterministic conditional divergences
Under deterministic conditional divergences, the appendix matches the exponent ceiling and describes flattened designs’ errors and horizon costs. Its closed-form horizon result requires additional constancy across time.
A.2 Deterministic conditional divergences
- Assumption A.8 makes each conditional normalizer deterministic; it holds for independent product measures, even when observations are not identically distributed, but fails when one-step divergences depend on the past.No constancy across time is required by A.8 itself.Source: §A.2 · p. 40
- For conditionally non-degenerate likelihood ratios with finite KL_n, c_{β,n} increases from 0 to KL_n while λ_n(β) decreases from KL_n to 0.The stated endpoint and monotonicity claims apply for β in (0,1).Source: §A.2 · p. 40
- Under A.8, the t-step ceiling equals the sum of the conditional ceilings and is approached by the conditional flattened designs as β decreases to 0.For the specified non-degenerate setting, the supremum is not attained by the betting sequence described in the theorem.Source: §A.2 · p. 41
- The exact error of a flattened design is expressed through whether the log-likelihood ratio G_n stays below its threshold B_n(β) through time t.The sequential error is P1(G_n≤B_n(β) for all n≤t).Source: §A.2 · p. 41
- The horizon-optimal tilt result carries over when A.8 holds and λ_n(β)=λ_1(β) for every n; without time constancy, the crossover and rate have no closed form.Extending the appendix results to β>1 would additionally require finite, P1-a.s. constant normalizers there.Source: §A.2 · pp. 41–42
- With conditional e-power at most c and conditional log-e-variable variance at most v, the sequential type-II error is at least 1−2v/(cℓ) for every t≤ℓ/(2c), when cℓ>2v.This is the stated universal horizon floor.Source: §A.2 · p. 42
- Under the stated variance and bounded-divergence conditions, the near-ceiling design has power at most 2β²v̄/(c̄_βℓ)=O(β) for t≤ℓ/(2c̄_β), a horizon range growing like β^-1.The passage requires the additional finite bound J on the essential suprema for this horizon range to be non-empty.Source: §A.2 · p. 42
A.3 A Stein-type converse
The appendix gives a Stein-type converse under an information-spectrum assumption and relates it to the i.i.d. case. It also notes that the corresponding non-identically distributed conditional KL terms need not be constant.
A.3 A Stein-type converse
- Under Assumption A.18, Proposition A.19 gives a bound on type-II error for every betting sequence and significance level.The i.i.d. model with finite KL divergence satisfies the assumption with D = KL(P0‖P1).Source: §A.3 · pp. 42–43
- In the i.i.d. model with finite KL(P0‖P1), the strong law supplies the information-spectrum limit used by the proposition.The appendix says Proposition A.19 then contains Proposition 4.15.Source: §A.3 · p. 43
- For non-identically distributed settings, the conditional KL terms need not be constant across steps.The appendix describes D as the limit of their time averages when the stated conditions hold.Source: §A.3 · p. 43
A.4 Where the time average goes
This appendix relates the ceiling to the time-averaged conditional KL divergence under stated regularity conditions. Whether that average converges, and to what, depends on the model.
A.4 Where the time average goes
- The proof uses a Doob decomposition of −log L_t, with predictable nondecreasing A_t and a P0-martingale M_t.Taking expectations gives E_P0[A_t] = E_P0[−log L_t].Source: §A.4 · p. 43
- With finite conditional variances and λ_n(0+) finite, the ceiling and the Stein exponent have one limit under the stated assumptions.The proposition identifies the ceiling's limiting value with D.Source: §A.4 · p. 43
- Under the proposition's bounded-variance condition, the martingale differences yield M_t/t → 0 P0-almost surely.This is the convergence step used to establish the information-spectrum condition.Source: §A.4 · p. 43
- In the i.i.d. model, KL_n is constant and equals KL(P0‖P1), so the ceiling's limit is that divergence.This is the i.i.d. regime described in the appendix.Source: §A.4 · p. 44
- For independent non-identically distributed observations, the average is a Cesàro mean that need not converge.The appendix gives an alternating sequence with liminf 1/3 and limsup 2/3, making the ceiling genuinely horizon-dependent.Source: §A.4 · p. 44
- In the stationary ergodic regime described, the ceiling converges to the same value as the normalized finite-time KL divergence.The appendix invokes the generalized Shannon–McMillan–Breiman theorem [Bar85],.Source: §A.4 · p. 44
- The appendix distinguishes the divergence directions: Assumption 2.3 constrains KL(P1‖P0), while equation (34) concerns KL(P0‖P1).It relates this asymmetry to the i.i.d. result in Theorem 4.29.Source: §A.4 · p. 44
A.5 Identifying the exponent, and power one
The appendix identifies the exponent for an i.i.d. process through a large-deviation argument, subject to additional regularity conditions. It also gives conditions under which positive conditional e-power ensures eventual rejection.
A.5 Identifying the exponent, and power one
- Under Assumptions A.24 and the additional hypotheses in Proposition A.25, the appendix identifies the exponent using the large-deviation rate function.The hypotheses include Ψ′(0) > 0 and interiority of 0 in the effective domain of Ψ⋆.Source: §A.5 · p. 44
- Assumption A.24 alone does not imply either additional hypothesis required by Proposition A.25.The appendix compares these requirements to positive e-power and the interiority condition of Proposition 2.23.Source: §A.5 · p. 44
- If conditional e-power is at least c > 0 and conditional log-wealth variance is bounded, rejection occurs eventually P1-almost surely for every α in (0, 1).This is Proposition A.26's power-one condition.Source: §A.5 · p. 44
B Results quoted from the literature
This appendix collects established inequalities and results used in the paper, with references rather than new proofs. The cited tools support the paper's bounds, asymptotic arguments, and variational characterization.
B Results quoted from the literature
- Ville's inequality bounds the probability that a nonnegative supermartingale starting at most one ever exceeds a threshold.The appendix cites Ville and [HRMS20, Lem. 1].Source: B Results quoted from the literature · p. 45
- Conditional Hölder supplies an inequality given a sub-σ-algebra, with equality characterized by proportional normalized powers.The appendix gives the conditional equality criterion and applies it to Theorem 4.9.Source: B Results quoted from the literature · p. 45
- Kolmogorov's maximal inequality and Cramér's theorem are included as tools for bounding maxima and establishing large deviations.The appendix cites [Bil95, Thm. 22.4],, and [DZ10, Thm. 2.2.3].Source: B Results quoted from the literature · p. 46
- The Chernoff–Stein lemma gives the asymptotic type-II error exponent for i.i.d. testing at bounded type-I error.The cited formulation includes a strong converse and permits α throughout (0, 1).Source: B Results quoted from the literature · p. 46
- Rényi divergence is nondecreasing in its order below one and converges to KL divergence as the order approaches one from below.The appendix identifies these results as underlying Lemma 4.5(ii) and Lemma 4.16.Source: B Results quoted from the literature · p. 46
- The variational formula expresses λ(β) as an infimum of weighted KL divergences and as a supremum over e-variables of a log-moment.The same Rβ attains both descriptions, and the bound persists under conditioning for every test supermartingale.Source: B Results quoted from the literature · pp. 46–47