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Everywhere irregularity of certain classes of random processes with stationary Gaussian increments

Published online by Cambridge University Press:  01 July 2016

P. L. Davies
P. L. Davies*
Affiliation:
University of Konstanz
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Abstract

This paper is concerned with everywhere local behaviour of certain classes of random processes which have stationary Gaussian increments. It is shown that for two classes of processes almost all the sample functions have the following property. The supremum of the increments in the neighbourhood of a point is everywhere of larger order than the standard deviation. For a third class of processes it is shown that the supremum is at least of the same order as the standard deviation.

Keywords

GAUSSIAN PROCESSESSTATIONARY INCREMENTSEVERYWHERE IRREGULARITYSUPREMUM OF INCREMENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 1 , April 1973 , pp. 103 - 118
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Belayev, Yu. K. (1960) Local properties of the sample functions of stationary Gaussian processes. Theor. Probability Appl. 5, 117–120.CrossRefGoogle Scholar
[2] Berman, S. M. (1969) Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137, 277–299.CrossRefGoogle Scholar
[3] Berman, S. M. (1969) Harmonic analysis of local times and sample functions of Gaussian processes. Trans. Amer. Math. Soc. 143, 269–281.CrossRefGoogle Scholar
[4] Berman, S. M. (1970) Gaussian processes with stationary increments: Local times and sample function properties. Ann. Math. Statist. 41, 1260–1272.CrossRefGoogle Scholar
[5] Berman, S. M. (1972) Gaussian sample functions: Uniform dimension and Holder conditions nowhere. Nagoya Math. J. 46, 63–86.CrossRefGoogle Scholar
[6] Dvoretzky, A. (1963) On the oscillation of the Brownian motion process. Israel J. Math. 1, 212–214.CrossRefGoogle Scholar
[7] Kahane, J. P. (1968) Some random series of functions. Heath Mathematical Monographs, Lexington, Massachusetts.Google Scholar
[8] Kawada, T. and Kôno, N. (1971) A remark on the nowhere differentiability of sample functions of Gaussian processes. (Communication from the referee).CrossRefGoogle Scholar
[9] Kôno, N. (1970) On the modulus of continuity of sample functions of Gaussian processes. J. Math. Kyoto Univ. 10–3, 493–536.Google Scholar
[10] Marcus, M. B. (1968) Gaussian processes with stationary increments possessing discontinuous sample paths. Pacific J. Math. 26, 149–157.CrossRefGoogle Scholar
[11] Ostrowski, A. M. (1952) Bounds for determinants. Proc. Amer. Math. Soc. 3, 26–30.CrossRefGoogle Scholar
[12] Yeh, J. (1967) Differentiability of sample functions in Gaussian processes. Proc. Amer. Math. Soc. 18, 105–108; correction 19 (1968), 843.CrossRefGoogle Scholar