The optimal betting wealth growth rate

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 $$\lim_{n \to \infty}n^{-1}\inf_{P \in (\mathscr P)^{n)^{{\circ\circ}}}} \mathrm{KL}(Q<sup>n,P)$$, where $${\mathscr P}ⁿ = {P^n: P \in \mathscr{P}}$$ and $(\mathscr {P}<sup>n)<sup>{\circ\circ}$ 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, $$\mathrm{KL}<em>{\inf}(Q,\mathscr{P}) := \inf</em>{P \in \mathscr P}\mathrm{KL}(Q,P)$$. 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.

• The paper presents an assumption-free characterization of the maximal exponential growth rate of wealth in sequential hypothesis testing using reverse information projections.
• It demonstrates that the optimal growth rate, which deviates from the classic KL-divergence measure, is achieved using blockwise test supermartingale constructions under composite nulls.
• The work provides necessary and sufficient conditions for power-one sequential tests and generalizes finite-horizon results to fully sequential i.i.d. settings.

The Optimal Betting Wealth Growth Rate: An Expert Analysis

Overview

The paper "The optimal betting wealth growth rate" (2604.25280) presents a comprehensive, assumption-free characterization of the maximal exponential growth rate of wealth in sequential hypothesis testing with Kelly-style betting, against general composite null hypotheses $P$ under potentially misspecified, arbitrary alternative distributions $Q$. Crucially, the work goes beyond classic pointwise Kullback-Leibler (KL) exponents, showing that the optimal achievable rate is given by a limit involving reverse information projections onto the bipolar of the null, and not always by the minimum KL-divergence $KL(Q,P)$. The results generalize prior finite-horizon and non-sequential treatments to the fully sequential i.i.d. setting, settle several open questions regarding minimax-optimal rates, and unify the attainability of power-one sequential tests, maximal expected log-wealth growth, and the geometric structure of the null.

Main Results

Bipolar Enlargement and Reverse Information Projection

The central finding is that the maximal per-sample asymptotic expected log-growth rate of any $e$-process (test supermartingale) under $Q$, betting against a composite null $\mathcal{P} \subseteq \mathcal{M}_1(X)$, is:

$$\lim_{n \to \infty} \frac{1}{n}\, \inf_{R \in (\mathcal{P}^n)^{\circ\circ}} KL(Q^n \| R)$$

where $(\mathcal{P}^n)^{\circ\circ}$ denotes the bipolar of the $n$-fold product null—the effective null indistinguishable by $n$-sample $Q$0-variables.

This quantity is, in general, strictly less than the naive $Q$1. Equality holds if and only if a $Q$2-lower-semicontinuity (w.l.s.c.) property of $Q$3 holds at $Q$4 (e.g., $Q$5 weakly compact). Otherwise, the actual growth rate can be zero even with $Q$6.

Supermartingales and Blockwise Structure

All attainable rates can be realized (to arbitrary precision) using blockwise test supermartingales. This construction leverages finite-horizon $Q$7-variables to create products over i.i.d. blocks, directly realizing the optimal per-block exponent given by the reverse information projection.

Left-constant interpolation of blockwise wealth between block times can be exploited by noncompliant stopping rules and is not, in general, an $Q$8-process. Thus, only block-aligned sample times can safely achieve the supremal rate without leaking risk.

Sequential Testability: Necessary and Sufficient Conditions

A necessary and sufficient condition for the existence of level-$Q$9 power-one sequential tests against $KL(Q,P)$0 (for composite null $KL(Q,P)$1) is that the $KL(Q,P)$2-step reverse KL to the bipolar,

$KL(Q,P)$3

is strictly positive for some $KL(Q,P)$4, and thus for all $KL(Q,P)$5 (by marginalization properties of the bipolar). This is a strictly stronger condition than $KL(Q,P)$6, answering open questions on testability under general composite nulls.

Extension to Composite Alternatives

Considering composite alternative classes $KL(Q,P)$7, the maximal achievable uniform expected log-wealth growth is given by the asymptotic robust game value

$KL(Q,P)$8

which, in general, is no longer given by the worst-case pointwise rate due to a minimax gap. This gap vanishes for finite alternatives or under sub-exponential covering conditions.

Structural Examples and Counterexamples

The paper delineates the tightness and necessity of structural conditions. For example, without weak-compactness or convexity, the limiting $KL(Q,P)$9-KL rate can drop to zero while the $e$0-distance remains strictly positive. If the null is finite (hence weakly compact but not convex), the minimax robust rate and the worst-case pointwise rate may disagree. These results sharply characterize the geometric and topological requirements for rate identity.

Implications

Practical Testing and Online Inference

The results establish that the classic Kelly-style betting exponent $e$1 only governs sequential evidence growth when $e$2 is well-behaved (weakly compact/convex), which is not generic in models specified by infinite or nonparametric nulls. In high-dimensional or adversarial settings, the limit is strictly slower—and in some cases, betting cannot yield exponential evidence at all.

Consequently, for practitioners, the construction of sequential tests and continuously monitored inference cannot blindly rely on KL-divergence rates when the null is irregular. Instead, practitioners must analyze the bipolar structure of their model class, or use the paper’s recipe for blockwise supermartingale construction at the least.

Theoretical Connections and Generalizations

This work fully sequentializes previous duality and information-projection principles developed for $e$3-variables in batch testing settings [Larsson et al.]. It establishes that the geometric bipolar enlargement is the appropriately minimal “closure” of the null for sequential testability, subsuming both measure-theoretic and topological nuances. The implications extend to test admissibility, optimal stopping, and anytime-valid inference, providing sharp, assumption-free generalizations of Neyman-Pearson theory for sequential composite nulls.

Notably, the linking of sequential testability, per-sample evidence rate, and existence of power-one tests through the same geometric criterion resolves long-standing gaps in sequential asymptotics.

Directions for Future Work

Several directions naturally arise:

• Explicit characterization for practical classes: Developing computable descriptions of the bipolar and the reverse projection for structured nulls (e.g., exponential families, Gaussian processes) would extend practical usage.
• Extensions beyond i.i.d. settings: Generalizing to dependent data or non-product settings, particularly for time series or exchangeable sequences, would broaden applicability.
• Connections to online learning/CDL: Since sequential evidence growth shares structure with regret-minimization and universal portfolio selection, further exploitation could link statistical testing and sequential decision theory.

Conclusion

This paper rigorously establishes that, under arbitrary composite nulls, the maximal achievable exponential rate of sequential evidence (wealth) growth is the normalized asymptotics of the reverse KL to the bipolar of the null, not the canonical KL-divergence. The results constitute a unification of geometric measure-theoretic characterization, sequential testability, and rate-optimal wealth growth, with robust extension to composite alternatives and without reliance on reference measures, convexity, or topological regularity. These findings should inform the design of rigorous, anytime-valid inference methods for both theoretical and applied sequential analysis.