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

We study the existence and the regularity of the local time of filtered white noises X={X(t),t[0,1]}. 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 : https://doi.org/10.5802/ambp.228
Classification : 60G15,  60G17
Mots clés : Local time, Local nondeterminism, Chung’s type law of iterated logarithm, Filtered white noises.
@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/}
}
Raby Guerbaz. Local time and related sample paths of filtered white noises. Annales Mathématiques Blaise Pascal, Tome 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 611857 | Zbl 0478.60059

[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 1968271 | Zbl 1023.60043

[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 | Article | MR 1629014 | Zbl 0932.60037

[4] S. M. Berman Gaussian processes with stationary increments: Local times and sample function properties, Ann. Math. Statist., Volume 41 (1970), pp. 1260-1272 | Article | MR 272035 | Zbl 0204.50501

[5] S. M. Berman Gaussian sample functions: uniform dimension and Hölder conditions nowhere, Nagoya Math. J., Volume 46 (1972), pp. 63-86 | MR 307320 | Zbl 0246.60038

[6] S. M. Berman Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J., Volume 23 (1973), pp. 69-94 | Article | MR 317397 | Zbl 0264.60024

[7] B. Boufoussi; M. Dozzi; R. Guerbaz On the local time of the multifractional Brownian motion, Stochastics and stochastic repports, Volume 78 (33-49), 2006 pages | MR 2219711 | Zbl 1124.60061

[8] W. Ehm Sample function properties of multi-parameter stable processes, Z. Wahrsch. verw. Gebiete, Volume 56 (195-228), 1981 pages | MR 618272 | Zbl 0471.60046

[9] D. Geman; J. Horowitz Occupation densities, Ann. of Probab., Volume 8 (1980), pp. 1 -67 | Article | MR 556414 | Zbl 0499.60081

[10] W. Li; Q. M. Shao Gaussian Processes : Inequalities, Small Ball Probabilities and Applications, Stochastic Processes: Theory and methods. Handbook of Statistics (K. Uhlenbeck, ed.), Volume 19, Edited by C.R. Rao and D. Shanbhag Elsevier, New York, 2001, pp. 533-598 | MR 1861734 | Zbl 0987.60053

[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 | Article | MR 1318191 | Zbl 0821.60043

[12] L.D. Pitt Local times for Gaussian vector fields, Indiana Univ. Math. J., Volume 27 (1978), pp. 309-330 | Article | MR 471055 | Zbl 0382.60055

[13] M. Priestley Evolutionary spectra and non stationary processes, J. Roy. Statist. Soc., Volume B 27 (1965), pp. 204-237 | MR 199886 | Zbl 0144.41001

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

Cité par document(s). Sources :