Local time and related sample paths of filtered white noises
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, p. 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
Keywords: Local time, Local nondeterminism, Chung’s type law of iterated logarithm, Filtered white noises.
@article{AMBP_2007__14_1_77_0,
     author = {Guerbaz, Raby},
     title = {Local time and related sample paths of filtered white noises},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {14},
     number = {1},
     year = {2007},
     pages = {77-91},
     doi = {10.5802/ambp.228},
     mrnumber = {2298805},
     zbl = {1144.60029},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2007__14_1_77_0}
}
Guerbaz, Raby. 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. http://www.numdam.org/item/AMBP_2007__14_1_77_0/

[1] Adler., R. Geometry of random fields, Wily, New York (1980) | MR 611857 | Zbl 0478.60059

[2] Benassi, A.; Cohen, Serge; Istas, Jacques Local self similarity and Hausdorff dimension, C.R.A.S., Tome Série I, tome 336 (2003), pp. 267-272 | MR 1968271 | Zbl 1023.60043

[3] Benassi, A.; Cohen, Serge; Istas, Jacques; Jaffard, S. Identification of filtered white noises, Stochastic Processes and their Applications, Tome 75 (1998), pp. 31-49 | Article | MR 1629014 | Zbl 0932.60037

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

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

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

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

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

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

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

[11] Monrad, D.; Rootzén, H. Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Th. Rel. Fields, Tome 101 (1995), pp. 173-192 | Article | MR 1318191 | Zbl 0821.60043

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

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

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