Asymptotic expansions in the CLT containing the last known moment in the main part (Q6187955)

Let \(\xi_1,\xi_2,\ldots\) be a sequence of independent and identically distributed random variables, each with zero mean, unit variance and a density function which is symmetric around zero. The authors give an asymptotic expansion for the density function of the normalized sum \((\xi_1+\cdots+\xi_n)n^{-1/2}\) in terms of moments and similar quantities, and the Hermite polynomials. An explicit bound is given on the remainder term in this expansion.