Lower functions and Chung's LILs of the generalized fractional Brownian motion

Abstract: Let $X:={X(t)}{t\ge0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$ \big{X(t)\big}{t\ge0}\overset{d}{=}\left{ \int_{\mathbb R} \left((t-u)+^{\alpha}-(-u)+^{\alpha} \right) |u|^{-\gamma} B(du) \right}_{t\ge0}, $$ with parameters $\gamma \in (0, 1/2)$ and $\alpha\in \left(-\frac12+ \gamma , \, \frac12+ \gamma \right)$. Continuing the studies of sample path properties of GFBM $X$ in Ichiba, Pang and Taqqu (2021) and Wang and Xiao (2021), we establish integral criteria for the lower functions of $X$ at $t=0$ and at infinity by modifying the arguments of Talagrand (1996). As a consequence of the integral criteria, we derive the Chung-type laws of the iterated logarithm of $X$ at the $t=0$ and at infinity, respectively. This solves a problem in Wang and Xiao (2021).