A generalisation of Ville’s inequality to monotonic lower bounds and thresholds
Year: 2025
Type: article
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 cleaner finite-time version of the law of the iterated logarithm.
Source: Statistics & Probability Letters
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,180
OpenAlex ID: W4415327120