We study numerically the stress distribution on the interface between two thick elastic media bounded by interfaces that include spatially correlated asperities. The interface roughness is described using the self-affine topography that is observed over a very wide range of scales from fractures to faults. We analyze the correlation properties of the normal stress distribution when the rough surfaces have been brought into full contact. The self affinity of the rough surfaces is described by a Hurst exponent H. We find that the normal stress field is also self affine, but with a Hurst exponent H-1. Fluctations of the normal stress are shown to be important, especially at local scales with anti-persistent correlations. ¿ 2000 American Geophysical Union