A nonuniform local limit theorem for Poisson binomial random variables via Stein’s method

We prove a nonuniform local limit theorem concerning approximation of the point probabilities $P(S=k)$ $P(S=k)$ , where $S={\sum }_{i=1}^n{X}_i$ $S=\sum_{i=1}^{n}X_{i}$ , and $X_1,\dots ,X_n$ $X_{1},\ldots ,X_{n}$ are independent Bernoulli random variables with possibly different success probabilities. Our proof uses Stein’s method, in particular, the zero bias transformation and concentration inequality approaches.