Local correlation dimension of multidimensional stochastic process

The computation of the local correlation dimension is a way for estimating the Hausdorff dimension of the image of multidimensional stochastic processes. It can be obtained from the asymptotic behavior of the self-intersection occupation measure around zero. In this paper, we replace the usual indicator function of the occupation measure by a Gaussian kernel. Hence, we obtain the consistency of the local correlation dimension for multivariate fractional Brownian motion. On the other hand, we show that any used norms on R^d give the same asymptotic behavior of the occupation measure. The use of a numerical procedure based on log−log least square estimator and Monte-Carlo experiments confirm the theoretical results and provide an efficient way of estimation of the Hausdorff dimension. In addition, we show that our proposed estimation method performs the univariate one on the estimation of the Hausdorff dimension.