On almost sure limit theorems for detecting long-range dependent, heavy-tailed processes
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Marcinkiewicz strong law of large numbers, n^-1/p∑_k=1^n (d_k- d)→ 0 almost surely with p∈(1,2), are developed for products d_k=∏_r=1^s x_k^(r), where the x_k^(r) = ∑_l=-∞^∞c_k-l^(r)ξ_l^(r) are two-sided linear process with coefficients {c_l^(r)}_l∈ℤ and i.i.d. zero-mean innovations {ξ_l^(r)}_l∈ℤ. The decay of the coefficients c_l^(r) as |l|→∞, can be slow enough for {x_k^(r)} to have long memory while {d_k} can have heavy tails. The long-range dependence and heavy tails for {d_k} 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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