A Generalisation of Ville's Inequality to Monotonic Lower Bounds and Thresholds

TL;DR

This paper extends Ville's inequality to monotonic bounds and thresholds, enabling more flexible anytime-valid methods for sequential analysis, and derives a finite-time law of the iterated logarithm.

Contribution

It generalizes Ville's inequality to monotonic curves, providing tight bounds and new tools for sequential probability analysis.

Findings

Bound is tight with an explicit supermartingale example

Derived a finite-time law of the iterated logarithm

Enables more flexible anytime-valid inference methods

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.