The Exact Ville Identity with Applications to E-Values
Abstract
Let Mn be a nonnegative supermartingale with M0=1, and let Tb be the first time Mn crosses b>1. Ville's inequality gives P(Tb<∞)≤ 1/b. We show that, at every finite time horizon N, 1=bP(Tb≤ N)+Vb,N+Db,N+Rb,N, where Vb,N is the expected overshoot at crossing, Db,N is the cumulative predictable decrease before crossing, and Rb,N is the expected value of paths that up to the time horizon N remained below the boundary b. The identity follows from telescoping and can be made horizon free without additional integrability assumption. It gives a formula for the slack in Ville's inequality and thus the conservatism of the threshold test Mn≥ b. For likelihood ratio martingales, the same terms also describe power: Rb,N is the probability, under the alternative, of not crossing by N, whereas bP(Tb≤ N)+Vb,N is the power. The identity also allows us to verify horizon-free guarantees from a finite computation: P(Tb<∞) lies between P(Tb≤ N) and P(Tb≤ N)+Rb,N/b, so a proposed threshold test Mn≥ bα can be shown to be of level α∈(0,1) for every time horizon based on computations at the single horizon N. We illustrate how Ville's identity yields sharper thresholds for Bernoulli and exponential likelihood-ratio tests, and for a conformal test of exchangeability, demonstrating Monte Carlo calibration under a composite, nonparametric null. Lastly, we also establish a continuous-time analogue of Ville's identity.
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BibTeX
@misc{pena2026exact,
title = {{The Exact Ville Identity with Applications to E-Values}},
author = {Victor H. de la Pena and Michael J. Klass and Victor K. de la Pena and Alexander Ly and Logan Neishloss and Fangyuan Lin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.04620},
eprint = {2607.04620},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04620},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Exact Ville Identity with Applications to E-Values
AU - Victor H. de la Pena
AU - Michael J. Klass
AU - Victor K. de la Pena
AU - Alexander Ly
AU - Logan Neishloss
AU - Fangyuan Lin
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2607.04620
UR - https://arxiv.org/abs/2607.04620
AB - Let M_n be a nonnegative supermartingale with M_0=1, and let T_b be the first time M_n crosses b>1. Ville's inequality gives P(T_b<∞)≤ 1/b. We show that, at every finite time horizon N, 1=bP(T_b≤ N)+V_b,N+D_b,N+R_b,N, where V_b,N is the expected overshoot at crossing, D_b,N is the cumulative predictable decrease before crossing, and R_b,N is the expected value of paths that up to the time horizon N remained below the boundary b. The identity follows from telescoping and can be made horizon free without additional integrability assumption. It gives a formula for the slack in Ville's inequality and thus the conservatism of the threshold test M_n≥ b. For likelihood ratio martingales, the same terms also describe power: R_b,N is the probability, under the alternative, of not crossing by N, whereas bP(T_b≤ N)+V_b,N is the power. The identity also allows us to verify horizon-free guarantees from a finite computation: P(T_b<∞) lies between P(T_b≤ N) and P(T_b≤ N)+R_b,N/b, so a proposed threshold test M_n≥ b_α can be shown to be of level α∈(0,1) for every time horizon based on computations at the single horizon N. We illustrate how Ville's identity yields sharper thresholds for Bernoulli and exponential likelihood-ratio tests, and for a conformal test of exchangeability, demonstrating Monte Carlo calibration under a composite, nonparametric null. Lastly, we also establish a continuous-time analogue of Ville's identity.
ER -