Surfaces aléatoires : approximation du temps local

Berzin, Corinne

Directeur de thèse : Dacunha-Castelle, Didier

Membres du jury : Dacunha-Castelle, Didier ; Bretagnolle, Jean ; Florens-Zmirou, Danielle ; Guyon, Xavier ; Labeyrie, Jacques

Thèses d'Orsay, no. 241 (1989) , 126 p.

Résumé

Let $X (t, ω) t \in ℝ^{d}, ω \in Ω, d \geq 2$ be a real stationary gaussian field, defined on a probability space $(Ω, 𝒜 P)$ . We look at the asymptotic behavior of a particular stochastic integral, with respect to the geometric measure of the $u$ -level sets, $u \in ℝ$ , of the regularized field, obtained by composition of a convolution of $X$ , say $X_{ϵ}$ , with a matrix normalization which contains part of the information contained in the spectral moments matrix of second order of $X_{ϵ}$ .

Under the condition that the covariance function is twice continuously differentiable out of a set of zero Lebesgue's measure, this functional converges in $L^{2} (Ω)$ to the local time of $X$ at the level $u$ . Furthermore, we give a bound for the speed of convergence.

Classification : 60G60, 60G15, 60G10, 60G57, 60J55, 65D10, 60F25, 60D05, 60E07, 60G50
Keywords: Random fields - Gaussian processes - Stationary processes - Random measures - Local time and additive functional - Smoothing, curve fitting - $L^p$-limit theorems - Geometric probability, stochastic geometry, random sets - Infinitely divisible distributions, stable distributions - Sums of independent random variables.

@phdthesis{BJHTUP11_1989__0241__A1_0,
     author = {Berzin, Corinne},
     title = {Surfaces al\'eatoires : approximation du temps local},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud Centre d'Orsay},
     number = {241},
     year = {1989},
     language = {fr},
     url = {http://archive.numdam.org/item/BJHTUP11_1989__0241__A1_0/}
}

TY  - BOOK
AU  - Berzin, Corinne
TI  - Surfaces aléatoires : approximation du temps local
T3  - Thèses d'Orsay
PY  - 1989
IS  - 241
PB  - Université de Paris-Sud Centre d'Orsay
UR  - http://archive.numdam.org/item/BJHTUP11_1989__0241__A1_0/
LA  - fr
ID  - BJHTUP11_1989__0241__A1_0
ER  -

%0 Book
%A Berzin, Corinne
%T Surfaces aléatoires : approximation du temps local
%S Thèses d'Orsay
%D 1989
%N 241
%I Université de Paris-Sud Centre d'Orsay
%U http://archive.numdam.org/item/BJHTUP11_1989__0241__A1_0/
%G fr
%F BJHTUP11_1989__0241__A1_0

Berzin, Corinne. Surfaces aléatoires : approximation du temps local. Thèses d'Orsay, no. 241 (1989), 126 p. http://numdam.org/item/BJHTUP11_1989__0241__A1_0/

Bibliographie
Cité par

[1] Azais, J.M: "Approximation du temps local des mouvements stables". C.R.Acad.Sci.Paris., t306, série I, 787-790 (1988). | Zbl

[2] Azais, J.M; Florens, D: "Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires". Prob. Th. Rel. Fields., 76,121-132 (1987). | MR | Zbl | DOI

[3] Belyaiev, Y: "Point processes and first passages problems". Proc. Sixth Berkeley Symp. Math. Stat. Prob., 3, 1-17 (1972). | MR | Zbl

[4] Berman, S.M: "Local times and sample functions properties of stationary gaussian processes". Trans. Amer. Math. Soc, 137,277-299 (1969). | MR | Zbl | DOI

[5] Berman, S.M: "Gaussian processes with stationary incréments: local times and sample functions properties". Ann. Math. Stat., 41, 1260-1272 (1970). | MR | Zbl | DOI

[6] Berman, S.M: "Local nondeterminism and local time of gaussian processes". Indiana Univ. Math. J., 23, 69-94 (1973). | MR | Zbl | DOI

[7] Berman, S.M: "Local nondeterminism and local times of general stochastic processes". Ann. Inst. Henri Poincaré.,19 , n°2, 189-207 (1983). | MR | Zbl | Numdam

[8] Berman, S.M: "Joint continuity of the local times of Markov processes". Z. Wahrs verw. Gebiete., 69, 37-46 (1985). | MR | Zbl | DOI

[9] Berzin, C: "Approximation du temps local des champs aléatoires gaussiens stationnaires par régularisation des trajectoires". C. R. Acad. Sci. Paris., t306, série I, 291-294 (1988). | MR | Zbl

[10] Cuzick, J: "Continuity of gaussian local times". The Ann. Prob., 10, n°3, 818-823 (1982). | MR | Zbl

[11] Davydov, Yu.A: "Local times for multiparameter random processes". Theory Prob. Appl., 23, n°3, 573-583 (1978). | Zbl | DOI

[12] Erdelyi, Magnus, Oberhettinger, Tricomi: "Higher transcendental functions". Bateman manuscript project, Vol 2, Mc Graw-Hill Book Compagny, INC (1953). | Zbl

[13] Feller, W: "An introduction to probability theory and it's applications". John Wiley ans Sons, Inc second corrected printed, Vol 2 (1966). | MR | Zbl

[14] Geman, D; Horowitz, J : "Occupation densities". The Ann. Prob., 8, n°1, 1-67 (1980). | MR | Zbl | DOI

[15] Gray, A; Mathews, Gb; Macrobert, T.M: "A treatise on Bessel functions and their applications to physics", Macmillan and Co, London, second édition (1952).

[16] Kahane, J.P; Salem, R: "Ensembles parfaits et séries trigonométriques". Hermann (1963). | MR | Zbl

[17] Kahane, J.P: "Ensembles aléatoires et dimensions". Cours au séminaire de L'Escurial, Prépub. Math. d'Orsay (1983). | Zbl | MR

[18] Kahane, J.P: "Mesures et dimensions". Fifth. Int. Congress. on Math Educ, held at Adélaïde (Australia) (1984).

[19] Massari, U; Miranda, M: "Minimal surfaces of codimension one". North-Holland, Mathematics Studies, Notas de Matemáticas (95), 91 (1984). | MR | Zbl

[20] Miranda, M: "Frontière minime". Mon. Math., 27, IMPA, Rio De Janeiro (1976).

[21] Miranda, M: "Medida geométrica e algumas aplicaçoes". IMPA, Rio De Janeiro (1979).

[22] Rogers, C.A: "Hausdorff measures". Camb. Univ. Press. (1970). | MR | Zbl

[23] Tricot, C: "Mesures et dimensions". Thèse d'état, Univ. Paris-Sud. Centre d'Orsay (1983).

[24] Widder, D: "The Laplace transform". Princeton University Press (1941). | MR | JFM | Zbl

[25] Wschebor, M: "Surfaces aléatoires: mesure géométrique des ensembles de niveau". Lect. Notes Math., n°1147, Springer-Verlag (1985). | MR | Zbl

[26] Yadrenko, M.L : "Spectral theory of random fiels".Optimization software, INC, Publications divisions, New-York (1983). | MR | Zbl

Adler, R: "The geometry of random fields". John Wiley and Sons (1980). | MR | Zbl

Cramer, H: "Métodos Matemáticos de estadistica". Aguilar (1960).

Cramer, H; Leadbetter, M.R: "Stationary and related stochastic processes". J. Wiley and Sons (1967). | Zbl

Doob, J.L: "Stochastic processes". J. Wiley and Sons (1953). | MR | Zbl

Marcus, M.B: "Gaussian processes with stationary incréments possessing discontinuous sample paths". Pacific. J . Math., 26, 149-157 (1968). | MR | Zbl | DOI

[1] J. M. Azais : "Conditions for convergence of number of crossings to the local time. Application to stable processes with independent increments and to gaussian processes". A paraître. | Zbl

[2] J.M. Azais et D. Florens-Zmirou : "Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires". A paraître. | Zbl

[3] S. M. Berman : "Local non determinism and local times of gaussian processes". Indiana Univ. Math. J., Vol 23, n°1, (1973) , pages 69-94 | MR | Zbl | DOI

[4] Y.A Davydov : "Local times for multiparameter random processes". Theory. Probab. Appl., Vol 23, n°3, (1978), pages 573-583. | DOI

[5] W. Ehm : "Sample properties of multiparameter stable processes". Z. Wahrsch. Verw. Gebiete, Vol 56, (1981), pages 195-228. | MR | Zbl | DOI

[6] M. Wschebor : "Surfaces aléatoires : mesure géométrique des ensembles de niveau". Lecture Notes in Math., Springer, Berlin-New York, (1985) | MR | Zbl

[1] E. M. Cabaña, Estimation of the spectral moments, by means of extrema, Reporte n° 84-08, Dto. de Matematicas y Ciencias de la Computacion, Universidad Simon Bolivar, Caracas, 1984.

[2] E. M. Cabaña, Affine Processes: A Test of Isotropy based on Level Sets, S.I.A.M. J. Appl. Math., 47, n° 4, 1987. | MR | Zbl

[3] E. M. Cabaña, Esperanzas para Intégrales sobre Conjuntos de Nivel Aleatorios, Actas II Congreso Latinoamericano de Probabilidades Y Estadistica Matematica (II CLAPEM), Caracas, 1985.

[4] I. Iribarren, Asymptotic behaviour of the integrals of a function on the level set of a mixing random field, Prob. and Math. Stat., 10, Fasc. 1, 1988. | MR | Zbl

[5] G. Lindgren, Spectral moments estimation by means of level crossing, Biometrika, 61, n°3, 1974, p. 401. | MR | Zbl | DOI

[6] M. S. Longuet-Higgins, The statistical analysis of a random moving surface, Phil. Trans. A., 249. 1957, p. 321-387. | MR | Zbl

[7] M. Wschebor, Surfaces aléatoires. Mesures géométriques des ensembles de niveau, Lecture Notes in Math., Springer-Verlag, n° 1147, 1985. | MR | Zbl