Strong local nondeterminism of spherical fractional Brownian motion

Abstract

Let B = B (x) , x ∈ S 2 be the fractional Brownian motion indexed by the unit sphere S 2 with index 0 < H ≤ , introduced by Istas [12]. We establish optimal upper and lower bounds for its angular power spectrum {d ℓ , ℓ = 0, 1, 2, . . .}, and then exploit its high-frequency behavior to establish the property of its strong local nondeterminism of B.

Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces

“…By Kolmogorov's zero-one law, we have P(E κ 1 ) = 0 or 1. This implies (18) with K = sup{κ 1 ≥ 0 : P(E κ 1 ) = 0}. Now we prove Theorem 2.…”

“…The analysis of sample path properties of random fields has been considered by many authors [12], [23], [33], [37], [38], but the index set of the random fields is typically restricted to be the Euclidean space R d . Recently the investigation of sample path properties of random fields over the unit sphere S d has been conducted by [17], [18], [19]. This paper is concerned with sample path properties of isotropic Gaussian random fields on a d-dimensional compact two-point homogeneous space M d .…”

Estimates for Distributions of Suprema of Spherical Random Fields

“…Distributions of suprema under more general non-Gaussian settings were treated in [3]. For spherical random fields the properties of sample paths, excursion sets and excursion probabilities were recently studied in [4,5,21,22,23,25,26]. Asymtotics for excursion probabilities P sup t∈S N X(t) ≥ u , as u → ∞, for a (locally) isotropic Gaussian random field X over N -dimensional unit sphere S N , were stated in [4,5] by applying different methods for the cases of smooth and non-smooth sample paths and appropriately adapting for the spherical case the techniques developed for the fields in R N .…”

Stable Processes with Stationary Increments Parameterized by Metric Spaces

“…Recent developments on fractional Lévy Brownian fields include for example [7,21] (indexed by S n ). More generally, in the spatial context a Gaussian process {G x } x∈M is often characterized by its variogram v(x, y) := E(G x − G y ) 2 (then stationary increments imply that v(x, y) is a function of d(x, y)), and there is already a huge literature on random fields from this aspect; see for example [3,42] for latest surveys on Gaussian random fields.…”

“…For the Karlin stable process in (3.3), in the Gaussian case it is a fractional Brownian motion with Hurst index H = β/2 ∈ (0, 1), and hence the path properties are well known. We expect then to be able to improve and obtain regularity results on sample paths (see [21] for M = S n ). In the stable case, however, even for M = R + it remains open whether the Karlin stable process (3.3) has a version in the space D for α ∈ [1, 2) (see [9,Remark 2]).…”