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Process convolutions yield random fields with flexible marginal distributionsand dependence beyond Gaussianity, but statistical inference is often hamperedby a lack of closed-form marginal distributions, and simulation-based inferencemay be prohibitively computer-intensive. We here remedy such issues through aclass of process convolutions based on smoothing a (d+1)-dimensional L\'evybasis with an indicator function kernel to construct a d-dimensionalconvolution process. Indicator kernels ensure univariate distributions in theL\'evy basis family, which provides a sound basis for interpretation,parametric modeling and statistical estimation. We propose a class of isotropicstationary convolution processes constructed through hypograph indicator setsdefined as the space between the curve (s,H(s)) of a spherical probabilitydensity function H and the plane (s,0). If H is radially decreasing, thecovariance is expressed through the univariate distribution function of H. Thebivariate joint tail behavior in such convolution processes is studied indetail. Simulation and modeling extensions beyond isotropic stationary spatialmodels are discussed, including latent process models. For statisticalinference of parametric models, we develop pairwise likelihood techniques andillustrate these on spatially indexed weed counts in the Bjertop data set, andon daily wind speed maxima observed over 30 stations in the Netherlands.

Spatial random field models based on L\'evy indicator convolutions