Explicit constants in the nonuniform local limit theorem for Poisson binomial random variables

In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities $P(S=k)$ P ( S = k ) when S=∑ᵢ₌₁ⁿXᵢ S = ∑ i = 1 n X i and X₁,X₂,… ,Xₙ X 1 , X 2 , … , X n are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein’s method and, in particular, the K -function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case.