Continuity in law of some additive functionals of bifractional Brownian motion

ABSTRACT

Let $B^{H,K}$ be a bifractional Brownain motion with indices $H\in (0,1)$ and $K\in (0,1]$. 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 $B^{H,K}$. 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 $B^{H,K}$. To prove our result, we use the decomposition in law of $B^{H,K}$ 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.

KEYWORDS:

• Anisotropic Besov space
• continuity in law
• limit theorem
• tightness
• bifractional brownian motion
• fractional brownian motion
• local time
• fractional derivative
• Slowly varying function

Acknowledgments

This paper was completed while the first author was visiting the Department of Statistics and Probability at Michigan State University (DSPMSU). This author would like to express his sincere thanks to the staff of DSPMSU for generous supported and hospitality, especially Prof. Yimin Xiao. The authors would like to thank the Editor and Associate Editor and two anonymous referees for their careful reading of the manuscript and useful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.