The increments of a sub-fractional Brownian motion

The sub-fractional Brownian motion {X
_H
(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 X
_H
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 X
_H
. The two above mentioned constants will play a key role in the statement of the results.