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

TL;DR

This paper investigates the lower classes and Chung's laws of the iterated logarithm for a generalized fractional Brownian motion, providing integral criteria and small ball probability estimates to understand its sample path behavior.

Contribution

It introduces new integral criteria for the lower classes of a generalized fractional Brownian motion and derives Chung-type laws of the iterated logarithm using small ball probability estimates.

Findings

Established integral criteria for lower classes at zero and infinity.

Derived Chung-type laws of the iterated logarithm for the process.

Highlighted the role of small ball probabilities in the proofs.

Abstract

Let $\{X(t)\}_{t\geqslant0}$ be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019): \begin{align*} \{X(t)\}_{t\ge0}\overset{d}{=}&\left\{ \int_{\mathbb R} \left((t-u)_+^{\alpha}-(-u)_+^{\alpha} \right) |u|^{-\gamma/2} B(du) \right\}_{t\ge0}, \end{align*} where $ \gamma\in [0,1), \ \ \alpha\in \left(-\frac12+\frac{\gamma}{2}, \ \frac12+\frac{\gamma}{2} \right)$ are constants. For any $\theta>0$, let \begin{align*} Y(t)=\frac{1}{\Gamma(\theta)}\int_0^t (t-u)^{\theta-1} X(u)du, \quad t\ge 0. \end{align*} Building upon the arguments of Talagrand (1996), we give integral criteria for the lower classes of $Y$ at $t=0$ and at infinity, respectively. As a consequence, we derive its Chung-type laws of the iterated logarithm. In the proofs, the small ball probability estimates play important roles.