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

Ran Wang () and Yimin Xiao ()
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Ran Wang: Wuhan University
Yimin Xiao: Michigan State University

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2442-2479

Abstract: Abstract The generalized fractional Brownian motion (GFBM) $$X:=\{X(t)\}_{t\ge 0}$$ X : = { X ( t ) } t ≥ 0 with parameters $$\gamma \in [0, 1)$$ γ ∈ [ 0 , 1 ) and $$\alpha \in \left( -\frac{1}{2}+\frac{\gamma }{2}, \, \frac{1}{2}+\frac{\gamma }{2} \right) $$ α ∈ - 1 2 + γ 2 , 1 2 + γ 2 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 $$H = \alpha -\frac{\gamma }{2}+\frac{1}{2} \in (0,1)$$ H = α - γ 2 + 1 2 ∈ ( 0 , 1 ) . When $$\gamma = 0$$ γ = 0 , X is the ordinary fractional Brownian motion. When $$\gamma \in (0, 1)$$ γ ∈ ( 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 $$t > 0$$ 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 $$\alpha +\frac{1}{2}$$ α + 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.

Keywords: Gaussian self-similar process; Generalized fractional Brownian motion; Exact uniform modulus of continuity; Small ball probability; Chung’s LIL; Tangent process; Lamperti’s transformation; 60G15; 60G17; 60G18; 60G22 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01148-8

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