no. 1

Weak convergence to fractional Brownian motion in some anisotropic Besov space
Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 1, pp. 1-17.

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},
     pages = {1--17},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {11},
     number = {1},
     year = {2004},
     doi = {10.5802/ambp.181},
     mrnumber = {2077234},
     zbl = {1077.60025},
     language = {en},
     url = {http://www.numdam.org/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) no. 1, pp. 1-17. doi : 10.5802/ambp.181. http://www.numdam.org/item/AMBP_2004__11_1_1_0/

[1] Bergh, J.; Löfström, J. Interpolation spaces. An introduction, Springer-Verlag, 1976 | MR 482275 | Zbl 0344.46071

[2] Billingsley, P. Convergence of Probability measures, Wiley, New York, 1968 | MR 233396 | Zbl 0172.21201

[3] Ciesielski, Z.; Keryacharian, G.; Roynette, B. Quelques espaces fonctionels associes à des processus Gaussiens, Studia Math, Volume 107 (1993) no. 2, pp. 171-204 | MR 1244574 | Zbl 0809.60004

[4] Csaki, E.; Shi, Z.; Yor, M. Fractional Brownian motions as ”higher-order” fractional derivatives of Brownian local times (Bolyai Society Mathematical Studies, to appear) | MR 1979974 | Zbl 1030.60073

[5] Jacod, J.; Shiryaev, A. N. Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenchaften, 288, Springer, Berlin, 1987 | MR 959133 | Zbl 0635.60021

[6] Kamont, A. Isomorphism of some anisotropic Besov and sequence spaces, Studia. Math, Volume 110 (1994) no. 2, pp. 169-189 | MR 1279990 | Zbl 0810.41010

[7] Kamont, A. On the fractional anisotropic Wiener field, Prob. and Math. Statistics, Volume 16 (1996) no. 1, pp. 85-98 | MR 1407935 | Zbl 0857.60046

[8] Kilbas, A. A.; Marichev, O. I.; Samko, S. G. Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publishers, 1993 | MR 1347689 | Zbl 0818.26003

[9] Lamperti, J. On convergence of stochastic processes, Trans. Amer. Math. Soc, Volume 104 (1962), pp. 430-435 | Article | MR 143245 | Zbl 0113.33502

[10] Marcus, M. B.; Rosen, J. p–variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments, Ann. Prob, Volume 20 (1992) no. 4, pp. 1685-1713 | Article | MR 1188038 | Zbl 0762.60069

[11] McKean, H. P. A Hölder condition for Brownian local time, J. Math. Kyoto Univ, Volume 1 (1962), pp. 195-201 | MR 146902 | Zbl 0121.13101

[12] Peetre, J. New thoughts on Besov spaces, Duke Univ. Math. Series, Durham, NC, 1976 | MR 461123 | Zbl 0356.46038

[13] Titchmarsh, E. C. Introduction to the theory of Fourier integrals, Second edition. Clarendon Press, Oxford, 1948

[14] Trotter, H. R. A property of Brownian motion paths, Illinois. J. Math, Volume 2 (1958), pp. 425-433 | MR 96311 | Zbl 0117.35502

[15] Yamada, T. On the fractional derivative of Brownian local times, J. Math. Kyoto Univ, Volume 25 (1985) no. 1, pp. 49-58 | MR 777245 | Zbl 0625.60090