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

TL;DR

This paper establishes Strassen's local law of the iterated logarithm for a generalized fractional Brownian motion, revealing how parameters and location influence the process's local behavior.

Contribution

It provides the first explicit local law of the iterated logarithm for this class of generalized fractional Brownian motions, extending previous global results.

Findings

Explicit characterization of local LIL for the process at any point

Demonstrates the influence of parameters lpha, mma, and location on local behavior

Differentiates from earlier global LIL results by providing local insights

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).