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

Let $X:=\{X(t){\}}_{t\ge 0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$\{X(t){\}}_{t\ge 0}{\{{\int }_{\mathbb{R}}((t-u{)}_{+}^{\alpha }-(-u{)}_{+}^{\alpha })\mathrm{\mid }u{\mathrm{\mid }}^{-\gamma }B(du)\}}_{t\ge 0},$$ 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).