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

Abstract

Let $\{X(t){\}}_{t⩾0}$ 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)_+^α-(-u)_+^α \right) |u|^{-γ/2} B(du) \right\}_{t\ge0}, \end{align*} where ${\hat{I}}^3\in [0,1),\hat{I}\pm \in \left(-\frac{1}{2}+\frac{\hat{I}}{³}2,\frac{1}{2}+\frac{\hat{I}}{³}2\right)$ are constants. For any $\hat{I}¸>0$, let \begin{align*} Y(t)=\frac{1}{Γ(θ)}\int_0^t (t-u)^{θ-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.

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