Dispersion points for linear sets and approximate moduli for some stochastic processes

Author:
Donald Geman

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be Lebesgue measurable; then has Lebesgue density 0 at the origin if and only if

**[1]**S. Berman,*Local nondeterminism and local times of Gaussian processes*, Indiana Univ. Math. J.**23**(1973), 69-94. MR**0317397 (47:5944)****[2]**H. Cramer and M. R. Leadbetter,*Stationary and related stochastic processes*, John Wiley and Sons, New York, 1967. MR**0217860 (36:949)****[3]**W. Feller,*An introduction to probability theory and it applications*, vol. II, John Wiley and Sons, New York, 1966. MR**0210154 (35:1048)****[4]**D. Geman and J. Horowitz,*Occupation densities*, Ann. Probability (to appear). MR**556414 (81b:60076)****[5]**D. Geman and J. Zinn,*On the increments of multidimensional random fields*, Ann. Probability**6**(1978), 151-158. MR**0461638 (57:1623)****[6]**C. Goffman and D. Waterman,*Approximately continuous transformations*, Proc. Amer. Math. Soc.**12**(1961), 116-121. MR**0120327 (22:11082)****[7]**K. Itô and H. P. McKean,*Diffusion processes and their sample paths*, Academic Press, New York, 1965. MR**0199891 (33:8031)****[8]**F. Knight,*Existence of small oscillations at zeros of Brownian motion*, Lecture Notes in Math., vol. 381, Springer, Berlin, 1974, pp. 134-149. MR**0373038 (51:9240)****[9]**M. B. Marcus,*Sample paths of Gaussian processes*, Northwestern University, Evanston, Ill., 1977.**[10]**V. Strassen,*An invariance principle for the law of the iterated logarithm*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**3**(1964), 211-226. MR**0175194 (30:5379)****1.**ll. J. B. Walsh,*Some topologies connected with Lebesgue measure*, Lecture Notes in Math., vol. 191, Springer, Berlin, 1971, pp. 290-310. MR**0375445 (51:11638)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
28A10,
26A15,
60G15,
60G17

Retrieve articles in all journals with MSC: 28A10, 26A15, 60G15, 60G17

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0536946-7

Keywords:
Lebesgue density,
approximate upper (lower) modulus,
approximate continuity,
Brownian motion,
Gaussian process,
scale-invariance

Article copyright:
© Copyright 1979
American Mathematical Society