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

Abstract

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

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