A local limit theorem assuming finite second moments via Stein’s method

Let [Formula: see text] be independent but not necessarily identically distributed integer-valued random variables and let [Formula: see text] Estimation of the point probabilities [Formula: see text] is a common problem that occurs in many applied domains such as insurance and biology. It is well known that under mild conditions the point probabilities may be approximated using a normal distribution. Local limit theorems seek to quantify the error in this approximation and typically give a smaller error than corresponding central limit theorems. In this paper we prove a quantitative local limit theorem under the minimal assumption that the summands of S have finite second moments. Previous studies either make stronger moment assumptions, do not provide simple explicit error bounds, or both. Our proof uses Stein’s method with truncation and the concentration inequality approach. As an example, we apply our result to sums of beta negative binomial random variables which are often used to model heavy-tailed count data.