The extended Ville's inequality for nonintegrable nonnegative supermartingales
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909340549513216 |
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| author | Wang, Hongjian Ramdas, Aaditya |
| author_facet | Wang, Hongjian Ramdas, Aaditya |
| contents | 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 $σ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01163 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The extended Ville's inequality for nonintegrable nonnegative supermartingales Wang, Hongjian Ramdas, Aaditya Probability Statistics Theory Methodology Machine Learning 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 $σ$-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. |
| title | The extended Ville's inequality for nonintegrable nonnegative supermartingales |
| topic | Probability Statistics Theory Methodology Machine Learning |
| url | https://arxiv.org/abs/2304.01163 |