Signed binomial approximation of binomial mixtures via differential calculus for linear operators

We obtain sharp estimates in signed binomial approximation of binomial mixtures with respect to the total variation distance. We provide closed form expressions for the leading terms, and show that the corresponding leading coefficients depend on the zeros of appropriate Krawtchouk polynomials. The special case of Pólya–Eggenberger distributions is discussed in detail. Our approach is based on a differential calculus for linear operators represented by stochastic processes, which allows us to give unified proofs.