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On almost sure limit theorems for detecting long-range dependent, heavy-tailed processes

Abstract

Marcinkiewicz strong law of large numbers, n1pk=1n(dkd)0 {n^{-\frac1p}}\sum_{k=1}^{n} (d_{k}- d)\rightarrow 0\ almost surely with p(1,2)p\in(1,2), are developed for products dk=r=1sxk(r)d_k=\prod_{r=1}^s x_k^{(r)}, where the xk(r)=l=ckl(r)ξl(r)x_k^{(r)} = \sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}\xi_l^{(r)} are two-sided linear process with coefficients {cl(r)}lZ\{c_l^{(r)}\}_{l\in \mathbb{Z}} and i.i.d. zero-mean innovations {ξl(r)}lZ\{\xi_l^{(r)}\}_{l\in \mathbb{Z}}. The decay of the coefficients cl(r)c_l^{(r)} as l|l|\to\infty, can be slow enough for {xk(r)}\{x_k^{(r)}\} to have long memory while {dk}\{d_k\} can have heavy tails. The long-range dependence and heavy tails for {dk}\{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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