For ARMA processes in which the white noise sequence has finite variance, predictors are usually determined by minimizing the expected squared error (see for example Fuller (1976) and Box and Jenkins (1976)). If the process is Gaussian, this procedure also minimizes the probabilities of large deviations . For processes with infinite variance , however, an alternative criterion for selection of a best predictor is needed. Alternative approaches which have been suggested include minimization of the expected absolute error and the pseudo spectral technique of Cambanis and Soltani (1982). Most criteria are complicated to use, and require precise knowledge of the distribution of the white noise. Therefor it would be extremely useful, in the infinite variance case, to have a predictor which is reasonably simple to compute, which does not require full knowledge of the distribution of and which (in a sense to be specified) minimizes the probabilities of large prediction errors. In order to predict unobserved values of a linear process with infinite variance, we introduce a linear predictors based on . When the linear process is driven by symmetric stable white noise one of the predictors minimizes the scale parameter of the error distribution. In the more general case when the driving white noise process has regularly varying tails with index , this predictor minimizes the size of the error tail probabilities. The procedure which minimizes the dispersion can be interpreted also as minimizing an appropriately defined -distance between the predictor and the random variable to be predicted. Using this method we derive explicitly the best linear predictor of in terms of for the process ARMA(1,1) and for the process AR(p). For higher order processes general analytic expressions are cumbersome, but we indicate how predictors can be determined numerically. We shall be concerned in this thesis with prediction of the causal stationary solution of the ARMA(p, q) equations, In this thesis we discuss a predictor, based on the natural criterion of minimizing error ‘dispersion’