A Generalisation of Ville's Inequality to Monotonic Lower Bounds and Thresholds
Wouter M. Koolen, Muriel Felipe Pérez-Ortiz, Tyron Lardy
TL;DR
This work generalizes Ville's inequality to settings where the lower bound and the crossing threshold are time-varying curves $-f(n)$ and $g(n)$, respectively. It provides a tight, explicit bound on the probability that a nonnegative supermartingale crosses the curve $g(n)$ while staying above $-f(n)$, with equality realized by a Floor Hugger martingale and a robust upper bound via an auxiliary nonnegative supermartingale. The authors additionally derive a continuous-time bound and demonstrate a finite-time Law of the Iterated Logarithm using a Gaussian-mixture construction, yielding explicit, time-uniform concentration guarantees. The discussion connects the extension to practical aspects such as one-sided LIL bounds and the interpretation in terms of multiple testing with subsidies, offering a practical framework for anytime-valid inference in sequential settings.
Abstract
Essentially all anytime-valid methods hinge on Ville's inequality to gain validity across time without incurring a union bound. Ville's inequality is a proper generalisation of Markov's inequality. It states that a non-negative supermartingale will only ever reach a multiple of its initial value with small probability. In the classic rendering both the lower bound (of zero) and the threshold are constant in time. We generalise both to monotonic curves. That is, we bound the probability that a supermartingale which remains above a given decreasing curve exceeds a given increasing threshold curve. We show our bound is tight by exhibiting a supermartingale for which the bound is an equality. Using our generalisation, we derive a clean finite-time version of the law of the iterated logarithm.