The optimal betting wealth growth rate

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.