Stationary self-similar random elds on the integer lattice

We establish several methods for constructing stationary self-similar random elds (ssf ’s) on the integer lattice by “random wavelet expansion”, which stands for representation of random elds by sums of randomly scaled and translated functions, or more generally, by composites of random functionals and deterministic wavelet expansion. To construct ssf ’s on the integer lattice, random wavelet expansion is applied to the indicator functions of unit cubes at integer sites. We demonstrate how to construct Gaussian, symmetric stable, and Poisson ssf ’s by random wavelet expansion with mother wavelets having compact support or non-compact support. We also generalize ssf ’s to stationary random elds which are invariant under independent scaling along di erent coordinate axes. Finally, we investigate the construction of ssf ’s by combining wavelet expansion and multiple stochastic integrals. c © 2001 Elsevier Science B.V. All rights reserved.