A law of large numbers for predicting several steps ahead

TL;DR

This paper establishes a generalized law of large numbers for multi-step ahead predictions of bounded random variables, extending classical results for martingales and optimizing decision-making processes.

Contribution

It introduces a new law of large numbers for predicting multiple steps ahead, improving precision and applicability in decision-making scenarios.

Findings

Law of large numbers holds for predicting N variables o(N) steps ahead

Generalizes standard law of large numbers for martingales

Applicable to decision making with bounded loss functions

Abstract

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.