In this paper, we propose a new interpretation of local limit theorems for univariate and multivariate distributions on lattices. Using elementary techniques, we show that – given a local limit theorem in the standard sense – the distributions are approximated well by the limit distribution, uniformly on intervals of possibly decaying length. We identify the maximally allowable decay speed of the interval lengths. Further, we show that for continuous distributions, the interval type local law holds without any decay speed restrictions on the interval lengths. We show that various examples fit within this framework, such as standardized sums of i.i.d. random vectors or correlated random vectors induced by multidimensional spin models from statistical mechanics.

在本文中，我们提出了对格上单变量和多变量分布的局部极限定理的新解释。使用基本技术，我们表明——给定标准意义上的局部极限定理——分布很好地近似于极限分布，均匀地分布在可能衰减长度的区间上。我们确定了间隔长度的最大允许衰减速度。此外，我们表明，对于连续分布，区间型局部定律在区间长度没有任何衰减速度限制的情况下成立。我们表明，各种示例都适合该框架，例如 iid 随机向量的标准化总和或由统计力学的多维自旋模型诱导的相关随机向量。