What question did this study set out to answer?

The aim is to establish a law of large numbers that allows for predicting multiple steps ahead with increased accuracy.

April 19, 2026Statistics & Probability Letters0 citationsOpen Access

A law of large numbers for predicting several steps ahead

Key Points

The aim is to establish a law of large numbers that allows for predicting multiple steps ahead with increased accuracy.
Proved a law of large numbers for predicting several steps ahead.
Examined uniformly bounded random variables.
Applied the findings to decision making with bounded loss functions.
Demonstrated the law holds for predicting N steps ahead with high precision.
Showed it generalizes the standard law of large numbers for martingales.
Indicated optimal predictions with limited loss impact over N steps.

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

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