Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes

2003 ◽  
Vol 65 (4) ◽  
pp. 317-329 ◽  
Author(s):  
Xavier Bardina ◽  
Maria Jolis ◽  
Ciprian A. Tudor
Keyword(s):  
Weak Convergence ◽  
Gaussian Processes ◽  
Brownian Sheet ◽  
Fractional Brownian Sheet ◽  
Two Parameter

scholarly journals On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Charles El-Nouty
Keyword(s):  
General Result ◽  
Gaussian Processes ◽  
Brownian Sheet ◽  
Random Vectors ◽  
Fractional Brownian Sheet ◽  
Integral Test ◽  
Small Ball Probability ◽  
Small Ball ◽  
Sufficiency Part ◽  
Necessity Part

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

On the Lamperti transform of the fractional Brownian sheet

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Marwa Khalil ◽  
Ciprian Tudor ◽  
Mounir Zili
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Gaussian Processes ◽  
Stationary Process ◽  
Similar Process ◽  
Brownian Sheet ◽  
Fractional Brownian Sheet ◽  
Self Similar ◽  
Strictly Stationary Process

AbstractIn 1962 Lamperti introduced a transformation that associates to every non-trivial self-similar process a strictly stationary process. This transform has been widely studied for Gaussian processes and in particular for fractional Brownian motion. Our aim is to analyze various properties of the Lamperti transform of the fractional Brownian sheet. We give the stochastic differential equation satisfied by this transform and we represent it as a series of independent Ornstein-Uhlenbeck sheets.

scholarly journals HERMITE VARIATIONS OF THE FRACTIONAL BROWNIAN SHEET

2012 ◽  
Vol 12 (03) ◽  
pp. 1150021 ◽  
Author(s):  
ANTHONY RÉVEILLAC ◽  
MICHAEL STAUCH ◽  
CIPRIAN A. TUDOR
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variable ◽  
Central Limit Theorems ◽  
Hurst Parameter ◽  
Brownian Sheet ◽  
Fractional Brownian Sheet ◽  
Two Parameter

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet Wα, βwith Hurst parameter (α, β) ∈ (0, 1)2. When [Formula: see text] or [Formula: see text] a central limit theorem holds for the renormalized Hermite variations of order q ≥ 2, while for [Formula: see text] we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time (1, 1).

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