  In this paper, we propose a new interpretation of local limit theorems for
univariate and multivariate distributions on lattices. 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.
