Local time and related sample paths of filtered white noises
Guerbaz, Raby1
1 Department of Mathematics Faculty of Sciences Semlalia Cady Ayyad University B.P. 2390 Marrakech MORROCO
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 : 10.5802/ambp.228
Classification : 60G15, 60G17
Mots-clés : Local time, Local nondeterminism, Chung’s type law of iterated logarithm, Filtered white noises.
Guerbaz, Raby 1

1 Department of Mathematics Faculty of Sciences Semlalia Cady Ayyad University B.P. 2390 Marrakech MORROCO 
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Guerbaz, Raby. 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. http://www.numdam.org/articles/10.5802/ambp.228/

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

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

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

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

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

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

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

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

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

[10] Li, W.; Shao, Q. M.; Uhlenbeck, K. 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] Monrad, D.; Rootzén, H. 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] Pitt, L.D. Local times for Gaussian vector fields, Indiana Univ. Math. J., Volume 27 (1978), pp. 309-330 | DOI | MR | Zbl

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

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

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