The classical central limit theorem states the uniform convergence of the distribution functions of the standardized sums of independent and identically distributed square integrable real-valued random variables to the standard normal distribution function. While first versions of the central limit theorem are already due to Moivre (1730) and Laplace (1812), a systematic study of this topic started at the beginning of the last century with the fundamental work of Lyapunov (1900, 1901). Meanwhile, extensions of the central limit theorem are available for a multitude of settings. This includes, e.g., Banach space valued random variables as well as substantial relaxations of the assumptions of independence and identical distributions. Furthermore, explicit error bounds are established and asymptotic expansions are employed to obtain better approximations.
Classical error estimates like the famous bound of Berry and Esseen are stated in terms of absolute moments of the random summands and therefore do not reflect a potential closeness of the distributions of the single random summands to a normal distribution. Non-classical approaches take this issue into account by providing error estimates based on, e.g., pseudomoments. The latter field of investigation was initiated by work of Zolotarev in the 1960's and is still in its infancy compared to the development of the classical theory. For example, non-classical error bounds for asymptotic expansions seem not to be available up to now.
In the present work we first establish a new non-classical bound for the central limit theorem error in the case of multidimensional random summands, which are not necessarily identically distributed. Up to now the most fargoing result in this general setting is due to Rotar (1977, 1978) and we improve upon his result w.r.t. the exponent of the pseudomoment.
Second, we study short asymptotic Edgeworth expansions in the case of real valued random summands, which are not necessarily identically distributed. Here we obtain a non-classical error bound, which to our best knowledge is the first result of this type in the literature.
pseudomoment, non-classical bound, central limit theorem, Edgeworth expansion