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

Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion: $$ \{X(t)\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ/2} B(du) \right\}_{t\ge0}, $$ with parameters $γ\in (0, 1)$ and $α\in \left(-1/2+ γ/2, \, 1/2+ γ/2 \right)$. This is a self-similar Gaussian process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes. The parameters $α$ and $γ$ determine the probabilistic and statistical properties of $X$. In particular, the parameter $γ$ introduces non-stationarity of the increments. In this paper, we prove Strassen's local law of the iterated logarithm of $X$ at any fixed point $t_0 \in (0, \infty)$, which describes explicitly the roles played by the parameters $α, γ$ and the location $t_0$. Our result is different from the previous Strassen's LIL for $X$ at infinity proved by Ichiba, Pang and Taqqu (2022).