We give some limit theorems for the occupation times of 1-dimensional Brownian motion in some anisotropic Besov space. Our results generalize those obtained by Csaki et al. [4] in continuous functions space.
@article{AMBP_2004__11_1_1_0, author = {Ait Ouahra, M.}, title = {Weak convergence to fractional Brownian motion in some anisotropic Besov space}, journal = {Annales math\'ematiques Blaise Pascal}, volume = {11}, year = {2004}, pages = {1-17}, doi = {10.5802/ambp.181}, zbl = {1077.60025}, mrnumber = {2077234}, language = {en}, url = {http://dml.mathdoc.fr/item/AMBP_2004__11_1_1_0} }
Ait Ouahra, M. Weak convergence to fractional Brownian motion in some anisotropic Besov space. Annales mathématiques Blaise Pascal, Tome 11 (2004) pp. 1-17. doi : 10.5802/ambp.181. http://gdmltest.u-ga.fr/item/AMBP_2004__11_1_1_0/
[1] Interpolation spaces. An introduction, Springer-Verlag (1976) | MR 482275 | Zbl 0344.46071
[2] Convergence of Probability measures, Wiley, New York (1968) | MR 233396 | Zbl 0172.21201
[3] Quelques espaces fonctionels associes à des processus Gaussiens, Studia Math, Tome 107 (1993) no. 2, pp. 171-204 | MR 1244574 | Zbl 0809.60004
[4] Fractional Brownian motions as ”higher-order” fractional derivatives of Brownian local times (Bolyai Society Mathematical Studies, to appear) | MR 1979974 | Zbl 1030.60073
[5] Limit theorems for stochastic processes, Springer, Berlin, Grundlehren der Mathematischen Wissenchaften, Tome 288 (1987) | MR 959133 | Zbl 0635.60021
[6] Isomorphism of some anisotropic Besov and sequence spaces, Studia. Math, Tome 110 (1994) no. 2, pp. 169-189 | MR 1279990 | Zbl 0810.41010
[7] On the fractional anisotropic Wiener field, Prob. and Math. Statistics, Tome 16 (1996) no. 1, pp. 85-98 | MR 1407935 | Zbl 0857.60046
[8] Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publishers (1993) | MR 1347689 | Zbl 0818.26003
[9] On convergence of stochastic processes, Trans. Amer. Math. Soc, Tome 104 (1962), pp. 430-435 | Article | MR 143245 | Zbl 0113.33502
[10] –variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments, Ann. Prob, Tome 20 (1992) no. 4, pp. 1685-1713 | Article | MR 1188038 | Zbl 0762.60069
[11] A Hölder condition for Brownian local time, J. Math. Kyoto Univ, Tome 1 (1962), pp. 195-201 | MR 146902 | Zbl 0121.13101
[12] New thoughts on Besov spaces, Duke Univ. Math. Series, Durham, NC (1976) | MR 461123 | Zbl 0356.46038
[13] Introduction to the theory of Fourier integrals, Second edition. Clarendon Press, Oxford (1948)
[14] A property of Brownian motion paths, Illinois. J. Math, Tome 2 (1958), pp. 425-433 | MR 96311 | Zbl 0117.35502
[15] On the fractional derivative of Brownian local times, J. Math. Kyoto Univ, Tome 25 (1985) no. 1, pp. 49-58 | MR 777245 | Zbl 0625.60090