Stein's method, heat kernel, and linear functions on the orthogonal groups

Combining Stein's method with heat kernel techniques, we study the function Tr(AO), where A is a fixed n by n real matrix over such that Tr(AA^t)=n, and O is from the Haar measure of the orthogonal group O(n,R). It is shown that the total variation distance of the random variable Tr(AO) to a standard normal random variable is bounded by 2 * squareroot(2) /(n-1), slightly improving the constant in a bound of Meckes, which was obtained by completely different methods.