Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

ABSTRACT

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett. 69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result.

KEYWORDS:

• Anisotropic Besov spaces
• limit theorem
• tightness
• continuity in law
• fractional Brownian motion
• sub-fractional Brownian motion
• local time
• fractional derivative
• slowly varying function

MATHEMATICS SUBJECT CLASSIFICATION:

• 2010
• 46E30
• 60B12

Acknowledgments

The authors would like to thank the Editor in chief and the anonymous referee for useful comments. This article was completed while M. Ait Ouahra was visiting the Department of Statistics and Probability at Michigan State University (DSPMSU). M. Ait Ouahra would like to express his sincere thanks to the staff of DSPMSU for generous support and hospitality, especially Prof. Yimin Xiao.

Funding

M. Ait Ouahra was supported by a Fulbright Visiting Scholar grant 2012–2013.