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The object of the present paper is to discuss the convergence of a sequence of probability distributions and some properties of the limit law, mainly by means of Fourier analysis. We first prove a fun...damental theorem on convergence of a sequence of distributions and then give some remarks about the Levy continuity theorem and further using it, we shall give another proofs of known theorem. Next we consider the mean $ m { f } $ of a characteristic function, $ f $, prove some theorems on the mean of the products of characteristic functions and we give the applications of it, especially we discuss the continuity of an infinite convolution. Lastly we shall prove the pure theorem in weak sense.show more
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