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

Abstract

Let $X:=\{X(t){\}}_{t\ge 0}$ be a generalized fractional Brownian motion given by $$\{X(t){\}}_{t\ge 0}\stackrel{d}{=}{\left\{{\int }_{\mathbb{R}}\left((t-u{)}_{+}^{\alpha }-(-u{)}_{+}^{\alpha }\right)|u{|}^{-\gamma /2}B(du)\right\}}_{t\ge 0},$$ 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).

19 pages; Keywords and Phrases: Gaussian self-similar process; Strassen's {law of the iterated logarithm}; generalized fractional Brownian motion; Lamperti's transformation