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

Abstract

Abstract We prove a nonuniform local limit theorem concerning approximation of the point probabilities P ( S = k ) $P(S=k)$ , where S = ∑ i = 1 n X i $S=\sum_{i=1}^{n}X_{i}$ , and X 1 , … , 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.

Keywords

• Poisson binomial random variable
• Local limit theorem
• Stein’s method
• Zero-bias transformation
• Concentration inequalities