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| Abstract |
In this paper we have introduced a perhaps new form of the contagious stochastic process, which may be recognized to be one of the simple and natural generalizations of the independent one and the PÓL...YA-EGGENBERGER’s scheme. The limit theorems we obtained hitherto are the generalizations of the classical and elementary ones in these fundamental schemes. It may be also to be remarked that we have incidentally obtained the construction-methods, which are simple in the standpoint of the theory of probability, of some classes of the infinitely divisible probability-laws, which are of the central importance in the realm of the independent scheme. In the case of the strong contagion we have obtained a class of distribution functions of PEARSON TYPE I, and in the case of the transition, we have reached the probability-law whose characteristic function is the product of those of the distribution functions of the PEARSON TYPE III. It is also to be remarked that our theorems show that the limiting probability-laws we obtained, except the Gaussian law, are such that their characteristic functions are the products of the limiting probability-laws which PÓLYA-EGGENBERGER established in their elementary scheme. The application of our limiting laws themselves will be given in another occasion. In conclusion the author wishes to point out that the problem of discussing the limiting theorems without the essential restrictions of our Assumptions I and II is still open to us, and that the problem suggests us that a more general form of the probability-law can be obtained as limiting theorems in our scheme.show more
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