Related papers: The extended Ville's inequality for nonintegrable nonnegative supermartingales

The extended Ville's inequality for nonintegrable nonnegative supermartingales

URL: http://arxiv.org/abs/2304.01163v3
Date: Tue, 08 Oct 2024 17:12:13 GMT
Title: The extended Ville's inequality for nonintegrable nonnegative supermartingales
Authors: Hongjian Wang, Aaditya Ramdas,
Abstract summary: We rigorously present an extended theory of nonnegative supermartingales requiring neither integrability nor finiteness. We derive a key maximal inequality foreshadowed by Robbins, which we call the extended Ville's inequality. We derive an extension of the method of mixtures, which applies to $sigma$-finite mixtures of our extended nonnegative supermartingales.
Score: 30.14855064043107
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Abstract: Following the initial work by Robbins, we rigorously present an extended theory of nonnegative supermartingales, requiring neither integrability nor finiteness. In particular, we derive a key maximal inequality foreshadowed by Robbins, which we call the extended Ville's inequality, that strengthens the classical Ville's inequality (for integrable nonnegative supermartingales), and also applies to our nonintegrable setting. We derive an extension of the method of mixtures, which applies to $\sigma$-finite mixtures of our extended nonnegative supermartingales. We present some implications of our theory for sequential statistics, such as the use of improper mixtures (priors) in deriving nonparametric confidence sequences and (extended) e-processes.

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