Summary
In 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 presented
In 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 rigorously
In 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.
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Additional information
Most of the results in the Part I of this paper have been given in the writer’s previous papers: On the convergence of classes of distributions, Ann. Inst. Statist. Math., Tokyo, 3, 7–15 (1951); A metrization of class-convergence of distributions, loc. cit., 5, 1–7 (1953); On the many-dimensional distribution functions, loc. cit., 5, 41–58 (1953).
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Takano, K. On some limit theorems of probability distributions. Ann Inst Stat Math 6, 37–113 (1954). https://doi.org/10.1007/BF02960515
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DOI: https://doi.org/10.1007/BF02960515