Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T20:13:00.099Z Has data issue: false hasContentIssue false

Gaussian Sample Functions: Uniform Dimension and Hölder Conditions Nowhere

Published online by Cambridge University Press:  22 January 2016

Simeon M. Berman*
Affiliation:
New York University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X(t), t≥0, be a real Gaussian process with mean 0, stationary increments, and a2(t) = E|X(t) - X(0)|2. Here dH(λ), for some bounded monotone H. We summarize the main results.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1972

References

[1] Beljaev, Ju.K., Continuity and Hölder’s conditions for sample functions of stationary Gaussian processes, Proc. Fourth Berkeley Sympos. Math. Stat. and Prob., Vol. 2: Contributions to probability theory, Univ. of California Press, Berkeley, Calif., 1961, pp. 2333.Google Scholar
[2] Berman, S.M., Some continuity properties of Brownian motion with the time parameter in Hilbert space, Trans. Amer. Math. Soc. 131 (1968), 182198.Google Scholar
[3] Berman, S.M., Local times and sample function properties of stationary Gaussian processes, Trans. Amer. Math. Soc. 137 (1969), 277299.Google Scholar
[4] Berman, S.M., Harmonic analysis of local times and sample functions of Gaussian processes, Trans. Amer. Math. Soc. 143 (1969), 269281.Google Scholar
[5] Berman, S.M., Gaussian processes with stationary increments: local times and sample function properties, Ann. Math. Statist. 41 (1970), 12601272.Google Scholar
[6] Billingsley, P., Convergence of Probability Measures, John Wiley, New York, 1968.Google Scholar
[7] Cramer, H., Mathematical Methods of Statistics, Princeton Univ. Press, Princeton, 1946.Google Scholar
[8] Cramer, H. and Leadbetter, M.R., Stationary and Related Stochastic Processes, John Wiley, New York, 1967.Google Scholar
[9] Dudley, R.M., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis 1 (1967), 290330.Google Scholar
[10] Ito, K. and McKean, H.P. Jr., Diffusion Processes and their Sample paths, Springer-Verlag, Berlin, 1965.Google Scholar
[11] Orey, S., Gaussian sample functions and the Hausdorff dimension of level crossings, Z. Wahrscheinlichkeitstheorie verw. Geb. 17 (1971), 3947.Google Scholar
[12] Ray, D.B., Sojurn times of diffusion processes, Illinois J. Math. 7 (1963),615630.Google Scholar
[13] Taylor, S.J., The a-dimensional measure of the graph and the set of zeros of a Brownian path, Proc. Cambridge Philos. Soc. 51 Part II (1955), 265274.Google Scholar