Exact uniform modulus of continuity and Chung’s LIL for the generalized fractional Brownian motion

Kavli Affiliate: Ran Wang

| First 5 Authors: Ran Wang, Yimin Xiao, , ,

| Summary:

The generalized fractional Brownian motion (GFBM) $X:={X(t)}_{tge0}$ with
parameters $gamma in [0, 1)$ and $alphain left(-frac12+frac{gamma}{2},
, frac12+frac{gamma}{2} right)$ 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 $H = alpha-frac{gamma}{2}+frac12 in(0,1)$.
When $gamma = 0$, $X$ is the ordinary fractional Brownian motion. For $gamma
in (0, 1)$, GFBM $X$ does not have stationary increments, and its sample path
properties such as H"older continuity, path
differentiability/non-differentiability, and the functional law of the iterated
logarithm have been investigated recently by Ichiba, Pang and Taqqu (2020).
They mainly focused on sample path properties that are described in terms of
the self-similarity index $H$ (e.g., LILs at the origin and at infinity).
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 law of iterated logarithm. Our results show that the local
regularity properties of GFBM $X$ away from the origin are determined by the
index $alpha +frac1 2$, 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 $H$.

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