Exact Uniform Modulus of Continuity and Chung’s LIL for the Generalized Fractional Brownian Motion

Abstract

The generalized fractional Brownian motion (GFBM) $X:=\{X(t)\}_{t\ge 0}$ with parameters $\gamma \in [0, 1)$ and $\alpha \in \left( -\frac{1}{2}+\frac{\gamma }{2}, \, \frac{1}{2}+\frac{\gamma }{2} \right)$ is a centered Gaussian H-self-similar process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes, where $H = \alpha -\frac{\gamma }{2}+\frac{1}{2} \in (0,1)$. When $\gamma = 0$, X is the ordinary fractional Brownian motion. When $\gamma \in (0, 1)$, GFBM X does not have stationary increments, and its sample path properties such as Hölder continuity, path differentiability/non-differentiability, and the functional law of the iterated logarithm (LIL) have been investigated recently by Ichiba et al. (J Theoret Probab 10.1007/s10959-020-01066-1, 2021). They mainly focused on sample path properties that are described in terms of the self-similarity index H (e.g., LILs at infinity or at the origin). In this paper, we further study the sample path properties of GFBM X and establish the exact uniform modulus of continuity, small ball probabilities, and Chung’s laws of iterated logarithm at any fixed point $t > 0$. Our results show that the local regularity properties away from the origin and fractal properties of GFBM X are determined by the index $\alpha +\frac{1}{2}$ instead of the self-similarity index H. This is in contrast with the properties of ordinary fractional Brownian motion whose local and asymptotic properties are determined by the single index H.