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.

翻译：本文刻画了在重复对一般独立同分布零假设$\mathscr{P}$进行凯利赌博时，数据由任意备择假设$Q$独立同分布生成情形下财富增长的最优速率。我们证明该速率等于$\lim_{n \to \infty}n^{-1}\inf_{P \in (\mathscr P)^n)^{\circ\circ}} \mathrm{KL}(Q^n,P)$，其中${\mathscr P}^n = \{P^n: P \in \mathscr{P}\}$，$(\mathscr {P}^n)^{\circ\circ}$表示其双极集。该速率可实现且不可超越。该量通常小于文献中更常见的量$\mathrm{KL}_{\inf}(Q,\mathscr{P}) := \inf_{P \in \mathscr P}\mathrm{KL}(Q,P)$。若$\mathrm{KL}_{\mathrm{inf}}(\cdot,\mathscr P)$在$Q$处为弱下半连续函数，我们证明两量相等；特别地，当$\mathscr P$为弱紧集时该性质成立。对于简单备择假设，我们首次给出功效为1的序贯检验存在的充要条件（无需对$\mathscr P, Q$施加假设）。我们还推导了针对复合备择假设$\mathscr Q$的最优最坏情况增长率。我们强调，简约过滤上的检验超鞅足以处理所有独立同分布检验问题，无需使用更一般的e过程。因此，我们将Larsson等人~\cite{larsson2025numeraire}的最新结果完整推广到序贯框架。