The optimal betting wealth growth rate

2026-04-28

AI Summary

This study investigates the optimal exponential rate of wealth growth in Kelly betting games when observations are drawn from an arbitrary alternative distribution \( Q \) while testing against a composite null hypothesis set \( \mathcal{P} \) of i.i.d. distributions. Leveraging tools from information geometry and duality theory, the authors derive for the first time an exact expression for this rate, showing it equals the minimal Kullback–Leibler (KL) divergence from \( Q \) to \( \mathcal{P} \), which is generally strictly smaller than the conventional \( \mathrm{KL}_{\inf} \). Under weak lower semicontinuity conditions, the two quantities coincide. Furthermore, by employing test supermartingales and bipolar set theory, the work establishes necessary and sufficient conditions for power-one sequential tests without distributional assumptions, fully extending Larsson et al.'s numeraire results to sequential composite hypotheses and demonstrating that test supermartingales suffice for all i.i.d. testing problems, yielding worst-case optimal growth rates against composite alternatives.

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.

Problem

• Kelly betting
• wealth growth rate
• KL divergence
• sequential testing
• composite hypothesis

Innovation

• Kelly betting
• KL divergence
• sequential testing
• test supermartingales
• bipolar set