Non-asymptotic analysis – general approach

Vladimir V. Ulyanov (Lomonosov Moscow State University)
Wednesday, July 13, 2016 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

First we give short review on recent approximation results for non-linear forms in independent random elements including asymptotic expansions. The errors of approximations could be described either in asymptotic way as an order of a remainder term with respect to number $n$ of random elements or in non-asymptotic form as a bound for remainder term with explicitly written dependence on $n$, moment characteristics and dimension $p$ of random elements or observations . The results are obtained by very different techniques such as expanding the characteristic function of the particular statistic or applying the expansions for convolutions. We show that for most of these expansions one could safely ignore the underlying probability model and its ingredients (like, e.g., proof of existence of limiting processes and its properties). Indeed, similar expansions and error bounds can be derived using a general scheme reflecting some (hidden) common features. This is the universal collective behavior caused by many independent asymptotically negligible variables influencing the distribution of a functional. The general approach for non-asymptotic analysis is described in terms of a sequence of smooth symmetric functions. Under weak conditions we prove the existence of a $<\!<$limit$>\!>$ function as a first order approximation together with $<\!<$Edgeworth-type$>\!>$ asymptotic expansions and error bounds of approximation. The applications of the results to the corresponding examples in probability and statistics, e.g. for high order $U$-statistics, Kolmogorov-Smirnov statistic and Free Central Limit theorem, will be discussed as well. In second part of the talk we consider high dimensional approximation problems when the dimension $p$ of random elements or observations is comparable with its number $n$.