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 $H$.

| Search Query: ArXiv Query: search_query=au:”Ran Wang”&id_list=&start=0&max_results=10