Nonclassical Error Bounds for Asymptotic Expansions in the Central Limit Theorem

We present the first nonclassical error bound for short asymptotic expansions in the central limit theorem in $R$ for the case of independent but not necessarily identically distributed random variables. In the case of independent identically distributed random variables under the Cramér condition, this bound provides the correct order of convergence $(1/n)$, and the leading term in the bound is determined by powers of the third and fourth pseudomoments of the random summands.