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A law of large numbers for predicting several steps ahead
Vladimir Vovk · 2025-08-25 · via math.PR updates on arXiv.org

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.

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