Continuity in law of some additive functionals of bifractional Brownian motion

Abstract

ABSTRACTLet be a bifractional Brownain motion with indices and . We prove the continuity in law, in some anisotropic Besov spaces, with respect to H and K. Our result generalizes those obtained by Jolis and Viles [Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theor. Probab. 20(2) (2007), pp. 133–152] of the fractional Brownian motion local time and gives a new result for the generalized fractional derivatives with kernel depending on slowly varying function of the local time of . Notice that their result was generalized by Wu and Xiao [Continuity in the Hurst index of the local times of anisotropic gaussian random fields, Stoch. Proc. Their Appl. 119 (2009), pp. 1823–1844] for wide class of anisotropic gaussian random fields satisfying some condition (A) which is not satisfied by . To prove our result, we use the decomposition in law of given by Lei and Nualart [A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), pp. 619–624]. Our result is also new in the space of continuous functions.