INVARIANCE PRINCIPLES FOR SELF-SIMILAR SET-INDEXED RANDOM FIELDS

For a stationary random field (X-j)(j is an element of Z)(d) and some measure mu on R-d, we consider the set-indexed weighted sum process S-n(A) = Sigma(j subset of Zd) mu(nA boolean AND R-j)(1/2) X-j, where R-j is the unit cube with lower corner j. We establish a general invariance principle under a p-stability assumption on the X-j's and an entropy condition on the class of sets A. The limit processes are self-similar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures mu and particular sets A, we show that these limits can be Levy (fractional) Brownian fields or (fractional) Brownian sheets.