The increments of a sub-fractional Brownian motion

Abstract

The sub-fractional Brownian motion {XH(t), t ≥ 0} with Hurst index 0 <; H <; 1, is an element of the QHASI class, a class of centered Gaussian processes which was introduced in 2015. It satisfies a specific assumption depending on the value of H. The study of the increments of XH consists mainly in investigating their limit properties under suitable conditions. The lim sup behavior was already investigated. It depends on two constants : the first one occurs in the quasi-helix property whereas the second one in the approximately stationary increments property. Here we investigate the lim inf behavior and assess the influence of the specific assumption of the process XH. The two above mentioned constants will play a key role in the statement of the results.