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: $$ {X(t)}{t\ge0}\overset{d}{=}\left{ \int_{\mathbb R} \left((t-u)+^{\alpha}-(-u)+^{\alpha} \right) |u|^{-\gamma/2} B(du) \right}_{t\ge0}, $$ with parameters $\gamma \in (0, 1)$ and $\alpha\in \left(-1/2+ \gamma/2, \, 1/2+ \gamma/2 \right)$. This is a self-similar Gaussian process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes. The parameters $\alpha$ and $\gamma$ determine the probabilistic and statistical properties of $X$. In particular, the parameter $\gamma$ 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 $\alpha, \gamma$ 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).