Interval type local limit theorems for lattice type random variables and distributions

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.