Weak convergence to fractional Brownian motion in some anisotropic Besov space

Abstract

We give some limit theorems for the occupation times of 1-dimensional Brownian motion in some anisotropic Besov space. Our results generalize those obtained by Csaki et al. [4] in continuous functions space.

Convergence in law for certain additive functionals of symmetric stable processes under strong topology

“…The similar result for fractional derivative of order 0 < γ < α−1 2 is given in Ait Ouahra [1]; more precisely we have the following regularity |D γ L .…”

“…Let {X t , t 0} be a symmetric stable process with an index α ∈ (1,2]. This means that X is a real-valued process with stationary independent increment such that,…”

“…where B H (·, ·) is a standard Brownian sheet independent of X and H (1) t (·) is a fractional derivative of order γ = 0 of Brownian local time (see definition below).…”

“…En notant (p t (x), x ∈ R, t 0) les densités de transition de X, on pose : c α = +∞ 0 (p t (0) − p t (1)) dt. Nous appelons le drap Brownien fractionnaire d'indice H ∈ [0, 1] un processus gaussien centré continu (B t (x); x ∈ R, t 0) de covariance EB s (x)B t (y) = (s ∧ t) 1 [7] que pour : y 1 , y 2 , . .…”

“…où B H (·, ·) est un drap Brownien independent de X et H (1) t (·) est la derivée fractionnaire d'ordre γ = 0 du temps local de X que nous allons définir dans la suite.…”