Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

Abstract

ABSTRACT. Let B H = {B H (t), t ∈ R N } be an (N, d)-fractional Brownian sheet with index

The fractional mixed fractional brownian motion and fractional brownian sheet

“…a centered Gaussian process with stationary increments satisfying B H1 (0) = 0, with probability 1, and IE(B H1 (s)) 2 and k 2H1 is a normalizing constant. We refer to Li and Shao [18] for further information on this field.…”

“…There is a huge literature on this process. We refer to Ayache et al [1], Ayache and Xiao [2], Belinsky and Linde [3], Kühn and Linde [14], Mason and Shi [20] and Xiao and Zhang [25] for further information on the FBS.…”

Sample Path Properties of Anisotropic Gaussian Random Fields

“…It has been proved in Ayache and Xiao (2005), Wu and Xiao (2007) that, for every ε ∈ (0, 1), fractional Brownian sheets satisfy Conditions (C1), (C2) and (C3) for all I ⊆ [ε, ∞) N .…”

“…It follows from Kahane (1985, p. 206) or Testard (1986, p.17) that if there exist positive and finite constants c 7,3 , c 7,4 and c 7,5 such that 12) where µ n = µ n [ε, 1] N denotes the total mass of µ n , then there is a subsequence of {µ n }, say {µ n k }, such that µ n k → µ in M + γ and µ is strictly positive with probability ≥ c 2 7,3 /(2c 7,4 ). In this case, it follows from (7.10) that µ has its support in L x almost surely.…”

“…5 Small ball probabilities 22 6 Hausdorff and packing dimensions of the range and graph 24 7 Hausdorff dimension of the level sets and hitting probabilities 35 8 Local times and their joint continuity 45 …”