| Let W be a random variable with mean 0 and variance 1. There always exists a random variable W* for which EWf(W) = Ef'(W*) for all bounded functions f with bounded derivatives f'. The distribution of W* is called W-zero-biased and is necessarily absolutely continuous. Using the equation EWf(W) = Ef'(W*) and Stein's method, one can easily obtain a simple error bound on the total variation distance between the distribution of W* and N(0,1). We use this result to study discretized normal approximation (in total variation) for sums of integer-valued random variables and obtain a Cramer-type moderate deviation for W for which W* can be coupled with W such that W - W* is bounded. We will also discuss the issues one faces in normal approximation for W in Kolmogrov metric. |