Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the continuity of local times


Author: Donald Geman
Journal: Proc. Amer. Math. Soc. 57 (1976), 321-326
MSC: Primary 60G17; Secondary 60G15
DOI: https://doi.org/10.1090/S0002-9939-1976-0420812-4
MathSciNet review: 0420812
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Several conditions are given for a stochastic process $ X(t)$ on $ [0,1]$ to have a local time which is continuous in its time parameter (for example, in the Gaussian case, the integrability of $ {[E{(X(t) - X(s))^2}]^{ - 1/2}}$ over the unit square). Furthermore, for any Borel function $ F$ on $ [0,1]$ with a continuous local time, the approximate limit of $ \vert F(s) - F(t)\vert/\vert s - t\vert$ as $ s \to t$ is infinite for a.e. $ t \in [0,1]$ and $ s\vert F(s) = F(t) $ is uncountable for a.e. $ t \in [0,1]$.


References [Enhancements On Off] (What's this?)

  • [1] S. M. Berman, Local times and sample function properties of stationary Gaussian processes, Trans. Amer. Math. Soc. 137 (1969), 277-299. MR 39 # 1009. MR 0239652 (39:1009)
  • [2] -, Harmonic analysis of local times and sample functions of Gaussian processes, Trans. Amer. Math. Soc. 143 (1969), 269-281. MR 40 #2155. MR 0248905 (40:2155)
  • [3] -, Gaussian processes with stationary increments: Local times and sample function properties, Ann. Math. Statist. 41 (1970), 1260-1272. MR 42 #6916. MR 0272035 (42:6916)
  • [4] D. Geman and J. Horowitz, Local times and supermartingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 29 (1974), 273-293. MR 50 # 11476. MR 0359021 (50:11476)
  • [5] S. Orey, Gaussian sample functions and the Hausdorff dimension of level crossings, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 39-47. MR 0279882 (43:5603)
  • [6] S. Saks, Theory of the integral, Dover, New York, 1964. MR 29 #4850. MR 0167578 (29:4850)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60G17, 60G15

Retrieve articles in all journals with MSC: 60G17, 60G15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0420812-4
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society