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A number of investigations have suggested the relevance of models exhibiting a sporadic or intermittent non-Gaussian character. We mention specifically some of the studies of R. Dwyer concerning the cracking of ocean ice. Here a class of simple models exhibiting such an intermittency are constructed. The second- and fourth-order spectra of the processes constructed are determined. The higher order cumulant spectra give some insight into the non-Gaussian character of the models.