Continuity in the Hurst index of the local times of anisotropic Gaussian random fields

Abstract

Let { { X H ( t ) , t ∈ R N } , H ∈ ( 0 , 1 ) N } be a family of ( N , d ) -anisotropic Gaussian random fields with generalized Hurst indices H = ( H 1 , … , H N ) ∈ ( 0 , 1 ) N . Under certain general conditions, we prove that the local time of { X H 0 ( t ) , t ∈ R N } is jointly continuous whenever ∑ ℓ = 1 N 1 / H ℓ 0 > d . Moreover we show that, when H approaches H 0 , the law of the local times of X H ( t ) converges weakly [in the space of continuous functions] to that of the local time of X H 0 . The latter theorem generalizes the result of [M. Jolis, N. Viles, Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theoret. Probab. 20 (2007) 133–152] for one-parameter fractional Brownian motions with values in R to a wide class of ( N , d ) -Gaussian random fields. The main argument of this paper relies on the recently developed sectorial local nondeterminism for anisotropic Gaussian random fields.