Stationary self-similar random fields on the integer lattice

We establish several methods for constructing stationary self-similar random fields (ssfʹs) on the integer lattice by “random wavelet expansion”, which stands for representation of random fields by sums of randomly scaled and translated functions, or more generally, by composites of random functionals and deterministic wavelet expansion. To construct ssfʹs 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 ssfʹs by random wavelet expansion with mother wavelets having compact support or non-compact support. We also generalize ssfʹs to stationary random fields which are invariant under independent scaling along different coordinate axes. Finally, we investigate the construction of ssfʹs by combining wavelet expansion and multiple stochastic integrals.