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

TL;DR

This paper investigates the sample path properties of generalized fractional Brownian motion (GFBM), establishing integral criteria for lower functions and deriving Chung-type laws of the iterated logarithm at zero and infinity.

Contribution

It provides new integral criteria for lower functions of GFBM and derives Chung-type laws of the iterated logarithm, advancing understanding of GFBM's sample path behavior.

Findings

Established integral criteria for lower functions of GFBM.

Derived Chung-type laws of the iterated logarithm at zero and infinity.

Solved an open problem in previous research on GFBM.

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).