Marcinkiewicz strong law of large numbers, almost surely with , are developed for products , where the are two-sided linear process with coefficients and i.i.d. zero-mean innovations . The decay of the coefficients as , can be slow enough for to have long memory while can have heavy tails. The long-range dependence and heavy tails for are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The results provide a means to estimate how much (if any) long-range dependence and heavy tails a sequential data set possesses, which is done for real financial data. All of the stocks we considered had some degree of heavy tails. The majority also had long-range dependence. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.
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