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On Some Asymptotic Properties Concerning Brownian Motion

Published online by Cambridge University Press:  22 January 2016

Tunekiti Sirao  and
Tosio Nisida
Tunekiti Sirao
Affiliation:
Mathematical Institute, Nagoya University and Institute for Mathematical Statistics, Kôbe University
Tosio Nisida
Affiliation:
Mathematical Institute, Nagoya University and Institute for Mathematical Statistics, Kôbe University
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About the behavior of brownian motion at time point oo, there are many results by P. Lévy, A. Khintchine etc. P. Lévy cited a theorem by A. Kolmogoroff as the most precise result in his famous book “Processus stochastiques et mouvement brownian” without proof. In this paper we shall prove this theorem, using the similar result about the random sequence by W. Feller, and then, applying the theorem of projective invariance by P. Lévy, we shall find also the behavior of brownian motion at time point 0 from the above theorem.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 4 , June 1952 , pp. 97 - 101
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1952

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) This condition is not any essential restriction, since A. Khintchine has already proved that e > 0 (<0) implies {(2+e) log2 t},1/2 ∈ u∞(Σ∞).

4) Loc. cit. 1).

5) A numerical monotonic sequence {øn} will be said to belong to the lower class 8 if with probability one the inequality

be satisfied for infinitely many n; on the contrary, if with probability one (A) be satisfied only for finitely many n, then {ø n } will be said to belong to the upper class u (the terminology due to P. Levy).