Optimal bounds in non-Gaussian limit theorems for \(U\)-statistics
It is known that \(U\)-statistics given by symmetric kernels of degree two, asymptotically behave like weighted quadratic forms. The limit variable turns out to be an infinite weighted sum of squares of i.i.d. standard normal random variables which produce the non-Gaussian limit. The present paper establishes an Edgeworth expansion for that type of \(U\)-statistics. Explicit bounds of order \(O(N^{-1})\) for the uniform error of the distribution functions are derived. The results are related to a special representation of \(U\)-statistics, which is based on a related series of independent Gaussian random variables. The approach is based on a spectral decomposition of the underlying kernel.
scientific article; zbMATH DE number 1359638
scientific article