Lower functions and Chungs LILs of the generalized fractional Brownian motion

Let $X:={X(t)}_{tge0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$ big{X(t)big}_{tge0}overset{d}{=}left{ int_{mathbb R} left((t-u)_+^{alpha}-(-u)_+^{alpha} right) |u|^{-gamma} B(du) right}_{tge0}, $$ with parameters $gamma in (0, 1/2)$ and $alphain left(-frac12+ gamma , , frac12+ gamma right)$. 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).