On the tradeoff between almost sure error tolerance and mean deviation frequency in martingale convergence
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arXiv
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| Format: | Preprint |
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2023
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| author | Estrada, Luisa F. Högele, Michael A. Steinicke, Alexander |
| author_facet | Estrada, Luisa F. Högele, Michael A. Steinicke, Alexander |
| contents | In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective modulus of convergence. For this purpose we generalize {an} elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies (error incidence), which was recently established by the authors. First we study martingale convergence in $L^2$, and in the setting of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences in two settings: uniformly bounded increments in $L^p$, $p\geq 2$, using the respective Baum-Katz-Stoica theorems, and uniformly bounded exponential moments with the help of the martingale estimates by Lesigne and Volný. We also present applications for the tradeoff for the multicolor generalized Pólya urn process, the Generalized Chinese restaurant process, statistical M-estimators, as well as the a.s.~excursion frequencies of the Galton-Watson branching process. Finally, we relate the tradeoff concept to the convergence in the Ky Fan metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09055 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the tradeoff between almost sure error tolerance and mean deviation frequency in martingale convergence Estrada, Luisa F. Högele, Michael A. Steinicke, Alexander Probability 60F15, 60E15, 60F10, 60G42, 60J80, 62F05 In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective modulus of convergence. For this purpose we generalize {an} elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies (error incidence), which was recently established by the authors. First we study martingale convergence in $L^2$, and in the setting of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences in two settings: uniformly bounded increments in $L^p$, $p\geq 2$, using the respective Baum-Katz-Stoica theorems, and uniformly bounded exponential moments with the help of the martingale estimates by Lesigne and Volný. We also present applications for the tradeoff for the multicolor generalized Pólya urn process, the Generalized Chinese restaurant process, statistical M-estimators, as well as the a.s.~excursion frequencies of the Galton-Watson branching process. Finally, we relate the tradeoff concept to the convergence in the Ky Fan metric. |
| title | On the tradeoff between almost sure error tolerance and mean deviation frequency in martingale convergence |
| topic | Probability 60F15, 60E15, 60F10, 60G42, 60J80, 62F05 |
| url | https://arxiv.org/abs/2310.09055 |