Fields with Exceptional Tangent Fields

The asymptotic self-similarity property describes the local structure of a random field. In this paper, we introduce a locally asymptotically self-similar second order field X_H, whose local structures at x=0 and at x0 are very far from each other. More precisely, whereas its tangent field at x0 is a Fractional Brownian Motion, its tangent field at x=0 is a Fractional Stable Motion. In addition, X_H, is asymptotically self-similar at infinity with a Gaussian field, which is not a Fractional Brownian Motion, as tangent field. Then, its trajectories regularity is studied. Finally, the Hausdorff dimension of its graphs is given.