 |
|
|
| | | | | |
|
|
|
|
|
The Convex Minorant of the Cauchy Process
|
Jean Bertoin, Universite Pierre et Marie Curie
|
Abstract
We determine the law of the convex minorant $(M_s,
sin [0,1])$ of a real-valued Cauchy process on the unit
time interval, in terms of the
gamma process. In particular, this enables us to deduce that
the paths of $M$ have a
continuous derivative, and that the support of
the Stieltjes measure $dM'$ has logarithmic
dimension one.
|
Full text: PDF
Pages: 51-55
Published on: January 20, 2000
|
Bibliography
-
R. Bass. Markov processes and convex minorants. Séminaire de
Probabilités XVIII: 28--41, 1984.
Math. Review
86d:60086 .
-
J. Bertoin. Lévy Processes. Cambridge University Press, 1996.
Math. Review 98e:60117 .
-
J. Bertoin.
Renewal theory for embedded regenerative sets.
Ann. Probab. , 27:1523--1535, 1999. Math. Review number not available.
- J. Bertoin and M. E. Caballero.
On the rate of growth of
subordinators with slowly varying Laplace
exponent. Séminaire de
Probabilités XXIX: 125--132, 1995.
Math. Review 98g:60134 .
-
J. Bertoin and J. Pitman.
Two coalescents derived from
the ranges of stable subordinators. To appear in
Electronic Journal of Probability.
Math. Review number not available.
-
E. Cinlar. Sunset over Brownistan. Stochastic Process. Appl.
40: 45-53, 1992.
Math. Review 17,273d .
-
M. Cranston, P. Hsu and P. March.
Smoothness of the convex hull of planar Brownian motion. Ann. Probab.
17: 144--150, 1989.
Math. Review 89m:60190 .
-
S. N. Evans.
On the Hausdorff dimension of Brownian cone points.
Math. Proc. Camb. Philos. Soc.
98: 343--353, 1985.
Math. Review 86j:60185 .
-
B. E. Fristedt and W. E. Pruitt.
Lower functions for increasing random
walks and subordinators.
Z. Wahrscheinlichkeitstheorie verw. Gebiete 18:
167--182, 1971.
Math. Review 45 #1250 .
-
I. I. Gihman and A. V. Skorohod .
The Theory of Stochastic Processes II.
Springer, Berlin, 1975.
Math. Review 51 #11656 .
-
P. Groeneboom.
The concave majorant of Brownian motion.
Ann. Probab.
11: 1016--1027, 1983.
Math. Review 85h:60119 .
-
M. Nagasawa and H. Tanaka.
Concave majorants of Lévy processes. Preprint (1999).
Math. Review number not available.
-
J. W. Pitman.
Remarks on the convex minorant of Brownian motion.
In: Seminar on Stochastic Processes
(Evanston, 1982), pp. 219--227, Progr. Probab. Statist., 5, Birkhäuser
Boston, Boston, Mass., 1983.
Math. Review
85f:60119 .
|
|
|
|
|
|
|
|
| | | | |
Electronic Communications in Probability. ISSN: 1083-589X |
|
Context