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ε-Entropy of the Brownian Motion with the Multi-Dimensional Spherical Parameter

Published online by Cambridge University Press:  22 January 2016

Yoshikazu Baba
Yoshikazu Baba*
Affiliation:
Kobe University of Commerce
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M.S. Pinsker [3] has given a general method of calculating the ε-entropy of a Gaussian process and obtained, for example, an exact proof of the estimate for the ε-entropy of the ordinary Brownian motion B(t), 0 ≦ t ≦ 1, which was presented without proof by A.N. Kolmogorov [1].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 31 - 40
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Kolmogorov, A.N.: Theory of the transmission of information, Amer. Math. Soc. Translations Ser. 2, 33 (1963), 291–321.Google Scholar
[2] Kolmogorov, A.N. and Tihomirov, V.M.: ε-entropy and ε-capacity of sets in functional spaces. Amer. Math. Soc. Translations Ser. 2, 17 (1961), 277–364.Google Scholar
[3] , M.C. 14 (1963) 59–100.Google Scholar
[4] McKean, H.P.: Brownian motion with a several-dimensional time, Theory Prob. Appl., 8–4 (1963), 335–354.CrossRefGoogle Scholar
[5] Erdélyi, A. and others: Higher transcendental functions I-III, McGraw-Hill Publ. New York (1953–1955).Google Scholar