Strassen's local law of the iterated logarithm for the generalized fractional Brownian motion

Abstract

Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion given by $$ \{X(t)\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^{\alpha}-(-u)_+^{\alpha} \right) |u|^{-\gamma/2} B(du) \right\}_{t\ge0}, $$ with parameters $\gamma \in (0, 1)$ and $\alpha\in \left(-1/2+ \gamma/2, \, 1/2+\gamma/2\right)$. This process was introduced by Pang and Taqqu (2019) as the scaling limit of a class of power-law shot noise processes. The parameters $\alpha$ and $\gamma$ govern the probabilistic and statistical properties of $X$. In particular, the parameter $\gamma$ breaks the stationarity of increments of $X$. In this paper, we establish Strassen's local law of the iterated logarithm for $X$ at a given point $t_0 \in (0, \infty)$. This result describes explicitly the roles played by the parameters $\alpha, \gamma$, and the location $t_0$. Our theorem differs from the earlier Strassen's {global law of the iterated logarithm} for $X$ proved by Ichiba, Pang and Taqqu (2022).