On some limit theorems of probability distributions

SummaryIn Part I the Khintchine’s uniqueness theorem for the class convergence of probability distributions is proved in a natural way by making use of inverses of distribution functions; its generalization to the multidimensional case is also proved; relations between different paired sequences of scaling constants and centering constants in limit problems of probability distributions are given; and the general method to determine scaling constants and centering constants is presentedIn Part II both an analytical derivation of the P. Lévy’s canonical form of the infinitely divisible multi-dimensional probability distribution and a necessary and sufficient condition, for the distributions of sums of asymptoticaly uniformly negligible independent multi-dimensional random variables to converge to a given infinitely divisible probability distribution are given. The logarithms of non-vanishing characteristic functions are treated rigorouslyIn Part III various versions of the multi-dimensional central limit theorem on sums of independent random variables are studied.The results in the last two parts are extensions of the known facts in the one-dimensional case to the multi-dimensional case.