2021/12/07 by Frédéri Viens, Viens, Frederi, Mohamed Erraoui +3
Computer Science · Economics, Econometrics and Finance · Physics and Astronomy · #28A78 #60G15 #60G17 #60J45 #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2112.03648
openalex publication_date 2021/12/07 · openalex created_date 2021/12/31 · openalex updated_date 2026/07/28
Let B be a d-dimensional Gaussian process on ℝ, where the component are independents copies of a scalar Gaussian process B0 on ℝ+ with a given general variance function γ2(r)=Var(B0(r)) and a canonical metric δ(t,s):=(𝔼(B0(t)-B0(s))2)1/2 which is commensurate with γ(t-s). We provide some general condition on γ so that for any Borel set E⊂ [0,1], the Hausdorff dimension of the image B(E) is constant a.s., and we explicit this constant. Also, we derive under some mild assumptions on γ an upper and lower bounds of ℙ\B(E)∩ F≠ ∅ \ in terms of the corresponding Hausdorff measure and capacity of E× F. Some upper and lower bounds for the essential supremum norm of the Hausdorff dimension of B(E)∩ F and E∩ B-1(F) are also given in terms of d and the corresponding Hausdorff dimensions of E× F, E, and F.