# The optimal betting wealth growth rate

arXiv: https://arxiv.org/abs/2604.25280 · HTML: https://mlsanity.com/paper/2604.25280/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 28 references that are in the map.
- First posted on arXiv 28 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds](https://mlsanity.com/paper/2504.19952/) (2025)
- [Asymptotically optimal sequential change detection for bounded means](https://mlsanity.com/paper/2602.05272/) (2026)
- [Asymptotically Optimal Sequential Testing with Markovian Data](https://mlsanity.com/paper/2602.17587/) (2026)
- [Power one sequential tests exist for weakly compact mathscr P against mathscr Pc](https://mlsanity.com/paper/2604.03218/) (2026)
- [Optimal Anytime-Valid Tests for Composite Nulls](https://mlsanity.com/paper/2512.20039/) (2025)

## Built on by (most cited first, 1 of 1)

- [A complete characterization of sequential testability and change detectability in i.i.d. models](https://mlsanity.com/paper/2609.05752/) (2026)

## Abstract (arXiv)

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 mathscrP, but the data are drawn i.i.d from an arbitrary alternative Q. We prove that it equals limn → ∞n-1inf_P ∈ (mathscr P)n)circcirc KL(Qn,P), where mathscr Pn = Pn: P ∈ mathscrP and (mathscr Pn)circcirc 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, KLinf(Q,mathscrP) := infP ∈ mathscr PKL(Q,P). If KLinf(·,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.~citelarsson2025numeraire to the sequential setting.