arxiv: 2604.25280 · v1 · submitted 2026-04-28 · 🧮 math.ST · math.PR· stat.ML· stat.TH

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The optimal betting wealth growth rate

Aaditya Ramdas, Ashwin Ram

Pith reviewed 2026-05-07 14:22 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.MLstat.TH

keywords Kelly bettingwealth growth rateKullback-Leibler divergencebipolarsequential hypothesis testingpower-one testse-processesi.i.d. null hypothesis

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The pith

The optimal long-run wealth growth rate when Kelly betting against a general i.i.d. null equals the limit of n inverse times the infimum KL divergence from the alternative product to the bipolar of the null products.

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

This paper finds the highest sustainable rate at which a bettor's wealth can grow in repeated Kelly bets when the data follow an arbitrary distribution Q but the bettor must protect against a composite i.i.d. null given by a set of distributions P. The rate equals the limit as the number of rounds n tends to infinity of one over n times the smallest Kullback-Leibler divergence between the n-fold product of Q and the bipolar of the n-fold products from P. This quantity is always at most the familiar infimum KL divergence over P, and equals it whenever the infimum KL map is weakly lower semicontinuous at Q or when P is weakly compact. The same limit also supplies the first necessary and sufficient condition for the existence of power-one sequential tests against simple alternatives, and it yields the optimal worst-case growth rate when the alternative itself is composite.

Core claim

We prove that the optimal wealth growth rate equals lim n→∞ n^{-1} inf_{P ∈ (𝒫^n)^{∘∘}} KL(Q^n,P), where this rate is achievable and one cannot do better. This quantity is in general smaller than KL_inf(Q,𝒫) := inf_{P ∈ 𝒫} KL(Q,P). If KL_inf(·,𝒫) is weakly lower semicontinuous at Q, the two quantities are equal; in particular this happens when 𝒫 is weakly compact. For simple alternatives we provide the first matching necessary and sufficient condition for when power-one sequential tests exist. We also derive the optimal worst-case growth rate against composite 𝒬. Test supermartingales on reduced filtrations suffice for all i.i.d. testing problems.

What carries the argument

The bipolar (∘∘) of the set of n-fold product measures drawn from the null hypothesis set 𝒫, which closes the set under the operations needed to obtain the tightest achievable KL bound on growth rate.

If this is right

The bipolar KL limit is always at most the ordinary infimum KL divergence and is strictly smaller in some cases.
Power-one tests exist if and only if the bipolar KL limit is positive when the alternative is simple.
An analogous bipolar construction gives the optimal guaranteed growth rate when the alternative is itself composite.
All i.i.d. testing problems can be solved using test supermartingales on reduced filtrations; more general e-processes are unnecessary.
The result extends previous numeraire arguments from fixed-sample to fully sequential settings.

Where Pith is reading between the lines

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

Practitioners facing composite nulls should compute the bipolar-adjusted rate rather than the naive infimum KL when sizing bets.
The same bipolar construction may tighten growth-rate bounds in other sequential decision problems that rely on divergence minimization.
Because reduced filtrations suffice, implementation of optimal betting strategies can avoid tracking the full history of past outcomes.

Load-bearing premise

Observations are drawn i.i.d. from the alternative Q while the null consists of i.i.d. distributions from 𝒫, which allows the entire problem to reduce to product measures and their bipolar.

What would settle it

For a concrete null set 𝒫 and alternative Q, compute the bipolar KL limit and then run repeated Kelly bets; if no strategy achieves growth approaching the limit, or if some strategy exceeds it, the optimality claim is false.

read the original abstract

This paper characterizes the best possible rate of growth of wealth in a Kelly betting game when repeatedly betting against a general i.i.d. null hypothesis $\mathscr{P}$, but the data are drawn i.i.d from an arbitrary alternative $Q$. We prove that it equals $\lim_{n \to \infty}n^{-1}\inf_{P \in (\mathscr P)^n)^{\circ\circ}} \mathrm{KL}(Q^n,P)$, where ${\mathscr P}^n = \{P^n: P \in \mathscr{P}\}$ and $(\mathscr {P}^n)^{\circ\circ}$ is its bipolar, i.e., this rate is achievable and one cannot do better. This quantity is in general smaller than a more popular quantity in the literature, $\mathrm{KL}_{\inf}(Q,\mathscr{P}) := \inf_{P \in \mathscr P}\mathrm{KL}(Q,P)$. If $\mathrm{KL}_{\mathrm{inf}}(\cdot,\mathscr P)$ is weakly lowersemicontinuous (w.l.s.c.) at $Q$, we show that the two quantities are equal; in particular, this happens when $\mathscr P$ is weakly compact. For simple alternatives, we provide the first matching necessary and sufficient condition for when power-one sequential tests exist (without assumptions on $\mathscr P, Q$). We also derive the optimal worst-case growth rate against composite $\mathscr Q$. We emphasize that test supermartingales on reduced filtrations suffice for all i.i.d. testing problems, and more general e-processes are not required. We thus completely generalize the recent results of Larsson et al.~\cite{larsson2025numeraire} to the sequential setting.

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

read the letter

This paper pins down the optimal asymptotic wealth growth rate in sequential Kelly betting against composite i.i.d. nulls as the limit of n^{-1} times inf KL over the bipolar of the product measures, and shows it is achievable. The rate can be strictly smaller than the usual KL_inf, with equality under weak lower semicontinuity at Q, which holds for weakly compact null sets. It also gives the first matching necessary and sufficient conditions for power-one tests when the alternative is simple, and derives the optimal worst-case rate against a composite alternative Q. The work generalizes the numeraire results of Larsson et al. to the sequential setting and notes that test supermartingales on reduced filtrations are enough for all i.i.d. problems, so broader e-processes are not needed. The math relies on standard convex analysis and KL properties for product measures, which fits the i.i.d. setup cleanly. The weak lower semicontinuity condition is stated explicitly and is a natural technical point rather than a hidden gap. The i.i.d. restriction is explicit, so the results do not claim to cover dependent data, but that is not a flaw within the paper's stated scope. This is for researchers in sequential statistics and betting-based inference who care about tight rates and test existence conditions. A reader working on e-processes or online learning will get direct value from the characterizations and the reduced-filtration emphasis. The claims are precise enough and the generalization is straightforward enough that it deserves a serious referee, even though the full proofs would need checking in review.

Referee Report

3 major / 3 minor

Summary. The paper characterizes the optimal asymptotic wealth growth rate for Kelly betting against a general i.i.d. composite null hypothesis 𝒫 when observations are i.i.d. from an arbitrary alternative Q. It proves this rate equals lim_{n→∞} n^{-1} inf_{P ∈ (𝒫^n)^{∘∘}} KL(Q^n, P), where (𝒫^n)^{∘∘} denotes the bipolar of the n-fold product measures; the rate is achievable via a test supermartingale and cannot be improved. The quantity is generally smaller than KL_inf(Q, 𝒫) := inf_{P∈𝒫} KL(Q,P), with equality when KL_inf(·,𝒫) is weakly lower semicontinuous at Q (e.g., when 𝒫 is weakly compact). For simple alternatives the paper supplies the first matching necessary and sufficient condition for existence of power-one sequential tests. It also derives the optimal worst-case growth rate against a composite alternative 𝒬 and shows that test supermartingales on reduced filtrations suffice for all i.i.d. testing problems, thereby generalizing the results of Larsson et al. (2025) to the sequential setting.

Significance. If the claimed characterizations and proofs hold, the work supplies a sharp, non-asymptotic-rate result for optimal betting wealth growth under composite i.i.d. nulls, together with explicit conditions under which the popular KL_inf quantity coincides with the bipolar construction. The emphasis on reduced-filtration supermartingales and the matching nec+suff condition for power-one tests are concrete advances over prior e-process literature. The paper also provides the first explicit optimal worst-case growth rate for composite alternatives. These contributions are load-bearing for the central claim and are supported by standard convex-analytic tools (bipolar theorem, KL divergence) applied to product measures.

major comments (3)

[§3, Theorem 1] §3 (Theorem 1 and its proof): the reduction of the sequential betting problem to the bipolar infimum over product measures relies on the i.i.d. assumption and the definition of the wealth process; the manuscript should explicitly verify that the supermartingale property is preserved under the reduced filtration and that no additional measurability conditions are required beyond those stated for the bipolar set.
[§4, Theorem 2] §4 (Theorem 2 on w.l.s.c.): the equality between the bipolar rate and KL_inf(Q,𝒫) is asserted when KL_inf(·,𝒫) is weakly lower semicontinuous at Q. The proof sketch invokes the bipolar theorem, but the manuscript should confirm that the weak topology on probability measures is the correct topology for the lower semicontinuity argument and that the infimum is attained or approximated appropriately.
[§5] §5 (power-one test characterization): the necessary and sufficient condition for existence of a power-one test is stated for simple alternatives. The argument appears to combine the growth-rate result with Ville’s inequality; the manuscript should make explicit how the test supermartingale is constructed from the optimal betting strategy and why the condition is both necessary and sufficient without further assumptions on 𝒫 or Q.

minor comments (3)

Notation: the manuscript alternates between script 𝒫 and plain P for the null class; a single consistent symbol would improve readability.
[Introduction] The abstract claims that “test supermartingales on reduced filtrations suffice”; this is a strong statement that should be highlighted with a short dedicated paragraph or remark in the introduction.
[Introduction] The generalization of Larsson et al. (2025) is mentioned but the precise points of departure (composite null, sequential vs. fixed-n, reduced filtration) could be listed explicitly in a comparison table or bullet list.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will incorporate clarifications in a revised version.

read point-by-point responses

Referee: [§3, Theorem 1] §3 (Theorem 1 and its proof): the reduction of the sequential betting problem to the bipolar infimum over product measures relies on the i.i.d. assumption and the definition of the wealth process; the manuscript should explicitly verify that the supermartingale property is preserved under the reduced filtration and that no additional measurability conditions are required beyond those stated for the bipolar set.

Authors: We agree that an explicit verification strengthens the exposition. In the revision we will insert a short lemma immediately after the statement of Theorem 1 showing that the wealth process, defined via the product of one-step betting factors that depend only on the current observation, remains a supermartingale when the filtration is reduced to the natural filtration generated by the i.i.d. sequence. The conditional-expectation property carries over directly because the betting factor at time t is measurable with respect to the t-th coordinate. Measurability of the resulting supermartingale follows from the fact that the bipolar set is defined via the weak topology on probability measures and the Radon–Nikodym construction used in the proof satisfies the standard Borel measurability requirements already implicit in the statement of the bipolar; no further conditions are imposed. revision: yes
Referee: [§4, Theorem 2] §4 (Theorem 2 on w.l.s.c.): the equality between the bipolar rate and KL_inf(Q,𝒫) is asserted when KL_inf(·,𝒫) is weakly lower semicontinuous at Q. The proof sketch invokes the bipolar theorem, but the manuscript should confirm that the weak topology on probability measures is the correct topology for the lower semicontinuity argument and that the infimum is attained or approximated appropriately.

Authors: We will expand the proof of Theorem 2 to make these points explicit. The weak topology is the correct topology because KL divergence is lower semicontinuous with respect to it (a standard fact recalled from Csiszár & Körner or Dembo & Zeitouni). Under the assumed weak lower semicontinuity of KL_inf(·,𝒫) at Q, any sequence of measures in the bipolar that approximates the infimum can be shown to converge weakly to an element whose KL divergence equals the bipolar infimum; the bipolar itself is weakly closed by the bipolar theorem. When 𝒫 is weakly compact the infimum is attained, which we will state as a corollary. These additions will be placed immediately after the current proof sketch. revision: yes
Referee: [§5] §5 (power-one test characterization): the necessary and sufficient condition for existence of a power-one test is stated for simple alternatives. The argument appears to combine the growth-rate result with Ville’s inequality; the manuscript should make explicit how the test supermartingale is constructed from the optimal betting strategy and why the condition is both necessary and sufficient without further assumptions on 𝒫 or Q.

Authors: We will add a dedicated paragraph in §5 that constructs the test supermartingale explicitly: for a simple alternative Q the optimal betting strategy yields the wealth process W_n = exp(n · r_n), where r_n is the normalized log-wealth increment whose almost-sure limit is the bipolar growth rate. This process is a supermartingale under every P in the bipolar (hence under the null) by the same argument used for Theorem 1. Ville’s inequality then implies that if the growth rate is positive, W_n → ∞ almost surely under Q, yielding a power-one test. Necessity follows because a non-positive growth rate precludes unbounded growth of any supermartingale under Q. The argument relies only on the i.i.d. structure and the definition of the bipolar; no extra assumptions on 𝒫 or Q are required. The revised text will contain this construction verbatim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via convex analysis

full rationale

The paper's central result equates the optimal asymptotic wealth growth rate to lim n^{-1} inf_{P in (P^n)^{oo}} KL(Q^n, P) and proves achievability via test supermartingales. This follows directly from the bipolar theorem applied to the convex set of product measures under the i.i.d. assumption, combined with standard properties of KL divergence; no step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The cited generalization of Larsson et al. is an extension of prior independent work rather than a justification for the main theorem, and the w.l.s.c. condition for equality with KL_inf is derived separately without circularity. The result is externally falsifiable via the stated assumptions on Q and P.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the i.i.d. assumption for both null and alternative together with standard convex analysis tools; no free parameters are fitted and no new entities are introduced.

axioms (2)

domain assumption Observations are independent and identically distributed under both the null hypothesis 𝒫 and the alternative Q.
This setup is stated explicitly in the abstract and enables the use of product measures Q^n and 𝒫^n in the KL expression.
standard math The bipolar operation from convex analysis is well-defined on the sets of probability measures and the KL divergence is lower semicontinuous where needed.
Invoked directly in the definition of the optimal rate as the infimum over (𝒫^n)^{∘∘}.

pith-pipeline@v0.9.0 · 5607 in / 1576 out tokens · 89870 ms · 2026-05-07T14:22:13.606210+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 20 canonical work pages

[1]

Shubhada Agrawal and Aaditya Ramdas.On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds. 2025. arXiv: 2504.19952 [math.ST].url: https://arxiv. org/abs/2504.19952

work page arXiv 2025
[2]

Sebastian Arnold and Eugenio Clerico.Optimal e-values for testing the mean of a bounded random variable against a composite alternative. 2026. arXiv: 2601.11347 [math.ST].url: https://arxiv.org/abs/2601.11347

work page arXiv 2026
[3]

Optimal gambling systems for favorable games

Leo Breiman. “Optimal gambling systems for favorable games”. In:Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability(1961)

1961
[4]

Log optimal portfolios

Thomas M Cover. “Log optimal portfolios”. In:Chapter in “Gambling Research: Gambling and Risk Taking,” Seventh International Conference. Vol. 4. 1987

1987
[5]

Universal Portfolios

Thomas M. Cover. “Universal Portfolios”. In:Mathematical Finance1.1 (1991), pp. 1–29. doi: 10.1111/j.1467-9965.1991.tb00002.x .url: https://doi.org/10.1111/j.1467- 9965.1991.tb00002.x

work page doi:10.1111/j.1467-9965.1991.tb00002.x 1991
[6]

Cover and Joy A

Thomas M. Cover and Joy A. Thomas.Elements of Information Theory. John Wiley & Sons, Inc., 2006.isbn: 9780471241959.doi: 10.1002/047174882X.url: https://onlinelibrary. wiley.com/doi/book/10.1002/047174882X

work page doi:10.1002/047174882x.url: 2006
[7]

Sanov Property, Generalized I-Projection and a Conditional Limit Theorem

Imre Csisz´ ar. “Sanov Property, Generalized I-Projection and a Conditional Limit Theorem”. In:The Annals of Probability12.3 (1984), pp. 768–793.doi: 10.1214/aop/1176993227.url: https://doi.org/10.1214/aop/1176993227

work page doi:10.1214/aop/1176993227.url: 1984
[8]

Doob.Stochastic Processes

Joseph L. Doob.Stochastic Processes. Wiley Publications in Statistics. New York: John Wiley & Sons, 1953

1953
[9]

Safe testing

Peter Gr¨ unwald, Rianne de Heide, and Wouter Koolen. “Safe testing”. In:Journal of the Royal Statistical Society Series B: Statistical Methodology86.5 (Nov. 2024), pp. 1091–1128. doi:10.1093/jrsssb/qkae011

work page doi:10.1093/jrsssb/qkae011 2024
[10]

Sample-optimal quantum process tomography with non-adaptive incoherent measurements

Peter Harremo¨ es, Tyron Lardy, and Peter Gr¨ unwald. “Universal Reverse Information Pro- jections and Optimal E-statistics”. In:2023 IEEE International Symposium on Information Theory (ISIT). Taipei, Taiwan, 2023, pp. 394–399.doi: 10.1109/ISIT54713.2023.10206494. 41

work page doi:10.1109/isit54713.2023.10206494 2023
[11]

Time-uniform, nonparametric, nonasymptotic confidence sequences

Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon. “Time-uniform, nonparametric, nonasymptotic confidence sequences”. In:The Annals of Statistics49.2 (Apr. 2021), pp. 1055–1080.doi:10.1214/20-AOS1176343406

work page doi:10.1214/20-aos1176343406 2021
[12]

23 KellyBench: A Benchmark for Long-Horizon Sequential Decision Making William L

J. L. Kelly. “A new interpretation of information rate”. In:The Bell System Technical Journal 35.4 (1956), pp. 917–926.doi:10.1002/j.1538-7305.1956.tb03809.x

work page doi:10.1002/j.1538-7305.1956.tb03809.x 1956
[13]

Anytime validity is free: inducing sequential tests

Nick W Koning and Sam van Meer. “Anytime validity is free: inducing sequential tests”. In: Journal of the Royal Statistical Society Series B: Statistical Methodology(2026), qkag050.doi: 10.1093/jrsssb/qkag050.url:https://doi.org/10.1093/jrsssb/qkag050

work page doi:10.1093/jrsssb/qkag050.url:https://doi.org/10.1093/jrsssb/qkag050 2026
[14]

The numeraire e-variable and reverse information projection

Martin Larsson, Aaditya Ramdas, and Johannes Ruf. “The numeraire e-variable and reverse information projection”. In:The Annals of Statistics53.3 (June 2025), pp. 1015–1043.doi: 10.1214/24-AOS2487

work page doi:10.1214/24-aos2487 2025
[15]

Martin Larsson, Johannes Ruf, and Aaditya Ramdas.A complete characterization of testable hypotheses. 2026. arXiv: 2601.05217 [math.ST].url: https://arxiv.org/abs/2601.05217

work page arXiv 2026
[16]

Springer Series in Statistics

Lucien Le Cam.Asymptotic methods in statistical decision theory. Springer Series in Statistics. New York, Berlin, Heidelberg: Springer-Verlag, 1986.isbn: 0-387-96307-3

1986
[17]

Ashwin Ram and Aaditya Ramdas.Asymptotically optimal sequential change detection for bounded means. 2026. arXiv: 2602.05272 [math.ST] .url: https://arxiv.org/abs/2602. 05272

work page arXiv 2026
[18]

Ashwin Ram and Aaditya Ramdas.Power one sequential tests exist for weakly compact P against P c. 2026. arXiv: 2604 . 03218 [math.ST].url: https : / / arxiv . org / abs / 2604 . 03218

2026
[19]

Game-Theoretic Statistics and Safe Anytime-Valid Inference

Aaditya Ramdas, Peter Gr¨ unwald, Vladimir Vovk, and Glenn Shafer. “Game-Theoretic Statistics and Safe Anytime-Valid Inference”. In:Statistical Science38.4 (Nov. 2023), pp. 576– 601.doi:10.1214/23-STS894

work page doi:10.1214/23-sts894 2023
[20]

2020.doi: 10.48550/arXiv.2009

Aaditya Ramdas, Johannes Ruf, Martin Larsson, and Wouter Koolen.Admissible anytime-valid sequential inference must rely on nonnegative martingales. 2020.doi: 10.48550/arXiv.2009. 03167. arXiv: 2009 . 03167 [math.ST].url: https : / / doi . org / 10 . 48550 / arXiv . 2009 . 03167

work page doi:10.48550/arxiv.2009 2020
[21]

Testing exchange- ability: Fork-convexity, supermartingales and e-processes

Aaditya Ramdas, Johannes Ruf, Martin Larsson, and Wouter M Koolen. “Testing exchange- ability: Fork-convexity, supermartingales and e-processes”. In:International Journal of Ap- proximate Reasoning141 (2022), pp. 83–109

2022
[22]

Statistical Methods Related to the Law of the Iterated Logarithm

Herbert Robbins. “Statistical Methods Related to the Law of the Iterated Logarithm”. In: The Annals of Mathematical Statistics41.5 (Oct. 1970), pp. 1397–1409.doi: 10.1214/aoms/ 1177696786

work page doi:10.1214/aoms/ 1970
[23]

Alhad Sethi, Kavali Sofia Sagar, Shubhada Agrawal, Debabrota Basu, and P. N. Karthik. Asymptotically Optimal Sequential Testing with Markovian Data. 2026. arXiv: 2602.17587 [math.ST].url:https://arxiv.org/abs/2602.17587

work page arXiv 2026
[24]

Journal of the Royal Statistical Society: Series A (Statistics in Society) , volume =

Glenn Shafer. “Testing by Betting: A Strategy for Statistical and Scientific Communication”. In:Journal of the Royal Statistical Society Series A: Statistics in Society184.2 (Apr. 2021), pp. 407–431.issn: 0964-1998.doi: 10 . 1111 / rssa . 12647. eprint: https : / / academic . oup . com / jrsssa / article - pdf / 184 / 2 / 407 / 49325712 / jrsssa _ 184 _ 2...

work page doi:10.1111/rssa.12647 2021
[25]

Test Martingales, Bayes Factors and p-Values

Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, and Vladimir Vovk. “Test Martingales, Bayes Factors and p-Values”. In:Statistical Science26.1 (Feb. 2011), pp. 84–101.doi: 10. 1214/10-STS347

2011
[26]

Shubhanshu Shekhar.Optimal Anytime-Valid Tests for Composite Nulls. 2025. arXiv: 2512. 20039 [math.ST].url:https://arxiv.org/abs/2512.20039

work page arXiv 2025
[27]

´Etude critique de la notion de collectif

Jean Ville. ´Etude critique de la notion de collectif. Paris: Gauthier-Villars, 1939.url: https: //eudml.org/doc/192893

1939
[28]

E-values: Calibration, combination and applications , volume=

Vladimir Vovk and Ruodu Wang. “E-values: Calibration, combination and applications”. In: Annals of Statistics49.3 (2021), pp. 1736–1754.doi:10.1214/20-AOS2020. A Omitted Proofs Proof of Lemma 3.3. Consider a particular k≥ 1 and abbreviate n := tk. By definition, Wtk is Ftk = σ(X1, . . . , Xn)-measurable. By the Doob-Dynkin lemma, we thus have a measurable...

work page doi:10.1214/20-aos2020 2021