Dispersion points for linear sets and approximate moduli for some stochastic processes
HTML articles powered by AMS MathViewer
- by Donald Geman PDF
- Trans. Amer. Math. Soc. 253 (1979), 257-272 Request permission
Abstract:
Let $\Gamma \in [0, 1]$ be Lebesgue measurable; then $\Gamma$ has Lebesgue density 0 at the origin if and only if \[ \int _\Gamma {{t^{ - 1}}\Psi ({t^{ - 1}} {\text {meas}}} \{ \Gamma \cap (0, t)\} ) dt < \infty \] for some continuous, strictly increasing function $\Psi (t) (0 \leqslant t \leqslant 1)$ with $\Psi (0) = 0$. This result is applied to the local growth of certain Gaussian (and other) proceses $\{ {X_t}, t \geqslant 0\}$ as follows: we find continuous, increasing functions $\phi (t)$ and $\eta (t) (t \geqslant 0)$ such that, with probability one, the set $\{ t:\eta (t) \leqslant \left | {{X_t} - {X_0}} \right | \leqslant \phi (t)\}$ has density 1 at the origin.References
- Simeon M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 (1973/74), 69–94. MR 317397, DOI 10.1512/iumj.1973.23.23006
- Harald Cramér and M. R. Leadbetter, Stationary and related stochastic processes. Sample function properties and their applications, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0217860
- William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
- Donald Geman and Joseph Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67. MR 556414
- Donald Geman and Joel Zinn, On the increments of multidimensional random fields, Ann. Probability 6 (1978), no. 1, 151–158. MR 461638, DOI 10.1214/aop/1176995620
- Casper Goffman and Daniel Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116–121. MR 120327, DOI 10.1090/S0002-9939-1961-0120327-6
- Kiyoshi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR 0199891
- Frank B. Knight, Existence of small oscillations at zeros of Brownian motion, Séminaire de Probabilités, VIII (Univ. Strasbourg, année universitaire 1972-1973), Lecture Notes in Math., Vol. 381, Springer, Berlin, 1974, pp. 134–149. MR 0373038 M. B. Marcus, Sample paths of Gaussian processes, Northwestern University, Evanston, Ill., 1977.
- V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 175194, DOI 10.1007/BF00534910
- J. B. Walsh, Some topologies connected with Lebesgue measure, Séminaire de Probabilités, V (Univ. Strasbourg, année universitaire 1969–1970), Lecture Notes in Math., Vol. 191, Springer, Berlin, 1971, pp. 290–310. MR 0375445
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 253 (1979), 257-272
- MSC: Primary 28A10; Secondary 26A15, 60G15, 60G17
- DOI: https://doi.org/10.1090/S0002-9947-1979-0536946-7
- MathSciNet review: 536946