Local limit theorems without assuming finite third moment

Abstract

One of the most fundamental probabilities is the probability at a particular point. The local limit theorem is the well-known theorem that estimates this probability. In this paper, we estimate this probability by the density function of normal distribution in the case of lattice integer-valued random variables. Our technique is the characteristic function method. We complete to relax the third moment condition of Siripraparat and Neammanee (J. Inequal. Appl. 2021:57, 2021) and the references therein and also obtain explicit constants of the error bound.