Local time and related sample paths of filtered white noises
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 77-91.

We study the existence and the regularity of the local time of filtered white noises $X=\left\{X\left(t\right),\phantom{\rule{4pt}{0ex}}t\in \left[0,1\right]\right\}$. We will also give Chung’s form of the law of iterated logarithm for $X$, this shows that the result on the Hölder regularity, with respect to time, of the local time is sharp.

DOI: 10.5802/ambp.228
Classification: 60G15,  60G17
Keywords: Local time, Local nondeterminism, Chung’s type law of iterated logarithm, Filtered white noises.
Raby Guerbaz 1

1 Department of Mathematics Faculty of Sciences Semlalia Cady Ayyad University B.P. 2390 Marrakech MORROCO
@article{AMBP_2007__14_1_77_0,
author = {Raby Guerbaz},
title = {Local time and related sample paths of filtered white noises},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {77--91},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {14},
number = {1},
year = {2007},
doi = {10.5802/ambp.228},
mrnumber = {2298805},
zbl = {1144.60029},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.228/}
}
TY  - JOUR
AU  - Raby Guerbaz
TI  - Local time and related sample paths of filtered white noises
JO  - Annales mathématiques Blaise Pascal
PY  - 2007
DA  - 2007///
SP  - 77
EP  - 91
VL  - 14
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.228/
UR  - https://www.ams.org/mathscinet-getitem?mr=2298805
UR  - https://zbmath.org/?q=an%3A1144.60029
UR  - https://doi.org/10.5802/ambp.228
DO  - 10.5802/ambp.228
LA  - en
ID  - AMBP_2007__14_1_77_0
ER  - 
%0 Journal Article
%A Raby Guerbaz
%T Local time and related sample paths of filtered white noises
%J Annales mathématiques Blaise Pascal
%D 2007
%P 77-91
%V 14
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.228
%R 10.5802/ambp.228
%G en
%F AMBP_2007__14_1_77_0
Raby Guerbaz. Local time and related sample paths of filtered white noises. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 77-91. doi : 10.5802/ambp.228. https://ambp.centre-mersenne.org/articles/10.5802/ambp.228/

[1] R. Adler. Geometry of random fields, Wily, New York, 1980 | MR | Zbl

[2] A. Benassi; Serge Cohen; Jacques Istas Local self similarity and Hausdorff dimension, C.R.A.S., Volume Série I, tome 336 (2003), pp. 267-272 | MR | Zbl

[3] A. Benassi; Serge Cohen; Jacques Istas; S. Jaffard Identification of filtered white noises, Stochastic Processes and their Applications, Volume 75 (1998), pp. 31-49 | DOI | MR | Zbl

[4] S. M. Berman Gaussian processes with stationary increments: Local times and sample function properties, Ann. Math. Statist., Volume 41 (1970), pp. 1260-1272 | DOI | MR | Zbl

[5] S. M. Berman Gaussian sample functions: uniform dimension and Hölder conditions nowhere, Nagoya Math. J., Volume 46 (1972), pp. 63-86 | MR | Zbl

[6] S. M. Berman Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J., Volume 23 (1973), pp. 69-94 | DOI | MR | Zbl

[7] B. Boufoussi; M. Dozzi; R. Guerbaz On the local time of the multifractional Brownian motion, Stochastics and stochastic repports, Volume 78 (33-49), pp. 2006 | MR | Zbl

[8] W. Ehm Sample function properties of multi-parameter stable processes, Z. Wahrsch. verw. Gebiete, Volume 56 (195-228), pp. 1981 | MR | Zbl

[9] D. Geman; J. Horowitz Occupation densities, Ann. of Probab., Volume 8 (1980), pp. 1 -67 | DOI | MR | Zbl

[10] W. Li; Q. M. Shao; K. Uhlenbeck Gaussian Processes : Inequalities, Small Ball Probabilities and Applications, Stochastic Processes: Theory and methods. Handbook of Statistics, Volume 19, Edited by C.R. Rao and D. Shanbhag Elsevier, New York, 2001, pp. 533-598 | MR | Zbl

[11] D. Monrad; H. Rootzén Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Th. Rel. Fields, Volume 101 (1995), pp. 173-192 | DOI | MR | Zbl

[12] L.D. Pitt Local times for Gaussian vector fields, Indiana Univ. Math. J., Volume 27 (1978), pp. 309-330 | DOI | MR | Zbl

[13] M. Priestley Evolutionary spectra and non stationary processes, J. Roy. Statist. Soc., Volume B 27 (1965), pp. 204-237 | MR | Zbl

[14] Y. Xiao Strong local nondeterminism and the sample path properties of Gaussian random fields (2005) (Preprint)

Cited by Sources: