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On some Asymptotic Properties Concerning Homogeneous Differential Processes

Published online by Cambridge University Press:  22 January 2016

Tunekiti Sirao
Tunekiti Sirao*
Affiliation:
Mathematical Institute, Nagoya University
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About the behaviour of brownian motion at time point ∞ there are many results by P. Levy and A. Khintchine etc. The method of W. Feller is applicable to a similar discussion about a homogeneous differential process. In this paper we shall study, applying his method, the properties of a homogeneous differential process.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 6 , October 1953 , pp. 95 - 107
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

1) Feller, W.: “The law of the iterated logarithm for identically distributed random variables.” Ann. of Math. vol. 47(1946)Google Scholar.

2) ω is the probability parameter.

3) The symbols E and V denote the expectation and the variance respectively.

4) This is not an essential restriction.

5) We may assume a ≦ 1 without losing generality.

6) denotes the convergence (divergence) of the integrals.

7) Knopp, K.: Theorie und Anwendung der Unendlichen Reihen, 2ed., Beriin, 1924, p. 127 CrossRefGoogle Scholar.

8) Kolmogoroff, A.: Grundbegriffe der Wahrscheinuchkeitsrechung, Berlin, 1933, p. 59 CrossRefGoogle Scholar.

9) [x] denotes the largest integer which does not exceed x.

10) loc. cit. 1).

11) loc. cit. 1).

12) loc. cit. 1).

13) Feller, W.: “The general form of the so-called law of the iterated logarithm.” Trans. Amer. Math. Soc. vol. 54(1943), pp. 373–402 Google Scholar.

14) loc. cit. 13).

15) loc. cit. 13).