The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent

In safe hypothesis testing with test supermartingales, Ville's inequality provides anytime-valid type-I error guarantees for every significance level α∈(0,1], if one rejects the null hypothesis whenever the wealth process first exceeds 1/α. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate mathbbEP1[log E], the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level c>0, every α and horizon t we construct e-variables of conditional e-power exactly c whose probability of not rejecting by t is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, Lambda(E)=sups≥0-log mathbbEP1[E-s], whose range is exactly determined: supE Lambda(E)=KL(P0|P1), the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, KL(P1|P0), and that one is attained, P0-a.s. uniquely, by the likelihood ratio R. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family Rβ/mathbbEP0[Rβ]: the ceiling as βdownarrow0, R at β=1. Which β is best is settled by the horizon, exactly: R is optimal at t=log(1/α)/KL(P1|P0) alone, beaten by sharpening (β>1) below it and by flattening above.