Higher-than-second-order statistics are used to derive and implement 2-D Gaussianity and linearity tests which validate the assumptions of random models which characterize texture analysis and synthesis in terms of first- and second-order statistics. The non-redundant region of the 2-D cumulant sequence and its Fourier transform, the bispectrum, are correctly defined and proven. General non-minimum phase and asymmetric non-causal AR (autoregressive) and ARMA (autoregressive moving average) models of textures are derived using cumulant statistics. Parameter estimators are obtained both by solving a set of linear equations and by minimizing a cumulant-matching criterion. Simulations on synthetic data are performed and the results of the higher-order analysis on real textures are reported.