A non-uniform rate of convergence in a local limit theorem

Abstract Let { Xn } be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in m th power for some integer m ≥ 1, then Z n = n −½ ( X 1 + … + X n ) has a pdf f n for all n ≥ m . Here we give a necessary and sufficient condition for sup as n . → ∞, where φ ( x ) is the standard normal pdf and M ( x ) is a non-decreasing function of x ≥ 0 such that M (0) > 0 and M ( x )/ x δ is non-increasing for 0 < δ ≤ 1.