On asymptotic expansions in the first uniform Kolmogorov theorem (Q1907902)

Kolmogorov proposed to approximate the \(n\)-fold convolution \(F^n\) of a distribution \(F\) not by a single infinitely divisible distribution but by the whole class. His `first uniform theorem' states that there exists a universal constant \(C_1\) with the property that for any distribution \(F\) and any \(n\) there is an infinitely divisible distribution \(D\) such that the uniform distance is bounded by \(|F^n - D |\leq C_1 n^{- 1/5}\). The bound was improved by Arak to the optimal rate \(n^{- 2/3}\). Le Cam found a similar estimate with an optimal rate \(n^{- 1/3}\) if \(D\) is restricted to all accompanying distributions. The present paper derives similar but more stringent estimates if \(D\) is in a more general class of distributions derived from the Bergström asymptotic expansion.