The book is devoted to the analysis of accuracy of the approximation of the distribution of the sum of independent but equally distributed stochastic elements of a Banach space by Gaussian distributions. The fundamental content is connected with an estimation of the convergence speed on sets with nonsmooth boundary in general Banach spaces. The method of study is founded on the approximation of the indicator of a set by smooth functions. Here, an estimation of the measure of an \(\epsilon\)- strip of the boundary of the set is necessary.
The book contains five chapters. In the first one, different facts of stochastic quantities in Banach spaces and their distributions are collected, in particular Gaussian ones. In the second chapter, questions connected with existence in Banach spaces of smooth functions are considered, in particular with the possibility of smooth approximation. The third chapter is devoted to the central limit theorem in Banach spaces, in particular in the spaces C(S) and \(c_ 0\). In the fourth chapter, questions connected with distributions of the norm of a Gaussian stochastic element in Banach space are considered and the estimate of the measure of an \(\epsilon\)-strip of the boundary of the set. In the space \(c_ 0\) an example of a Gaussian element with unbounded distribution is constructed.
The fundamental results are obtained in the fifth chapter. An estimate of order \(O(n^{-1/6})\) is established and it is shown, that in the general case it can not be improved. For Hilbert spaces an estimation of order \(O(n^{-1/2})\) is established.