标题： 多个步骤向前预测的大数定律 显示英文标题

标题： A law of large numbers for predicting several steps ahead

摘要： 本注释证明了一个预测多步 ahead 的大数定律，对于均匀有界随机变量的情况，该定律推广了鞅的标准大数定律；标准的大数定律对应于预测一步 ahead。 其主要结果表明，对于预测$N$个均匀有界随机变量$o(N)$步 ahead，大数定律成立，但其更加精确，在某些方面是最佳的。 这个大数定律被应用于一个具有有界损失函数的决策问题，该函数限制每个决策的影响到$o(N)$步。 显示英文摘要

摘要： 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.