A law of large numbers for predicting several steps ahead
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908955969585152 |
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| author | Vovk, Vladimir |
| author_facet | Vovk, Vladimir |
| contents | This note proves a law of large numbers for predicting several steps ahead, which, in the case of uniformly bounded random variables, generalizes the standard law of large numbers for martingales; the standard law of large numbers corresponds to predicting one step ahead. Its main result shows that the law of large numbers holds for predicting $N$ uniformly bounded random variables $o(N)$ steps ahead, but it is much more precise and in some respects optimal. This law of large numbers is applied to a problem of decision making with a bounded loss function limiting the impact of each decision to $o(N)$ steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17507 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A law of large numbers for predicting several steps ahead Vovk, Vladimir Probability 60F05, 60G42 (Primary) 60F10, 60G25, 91B06 (Secondary) This note proves a law of large numbers for predicting several steps ahead, which, in the case of uniformly bounded random variables, generalizes the standard law of large numbers for martingales; the standard law of large numbers corresponds to predicting one step ahead. Its main result shows that the law of large numbers holds for predicting $N$ uniformly bounded random variables $o(N)$ steps ahead, but it is much more precise and in some respects optimal. This law of large numbers is applied to a problem of decision making with a bounded loss function limiting the impact of each decision to $o(N)$ steps. |
| title | A law of large numbers for predicting several steps ahead |
| topic | Probability 60F05, 60G42 (Primary) 60F10, 60G25, 91B06 (Secondary) |
| url | https://arxiv.org/abs/2508.17507 |