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

12 July 2020
M. Kouritzin
Sounak Paul University of Alberta
ArXiv (abs)PDFHTML
Abstract

Marcinkiewicz strong law of large numbers, n−1p∑k=1n(dk−d)→0 {n^{-\frac1p}}\sum_{k=1}^{n} (d_{k}- d)\rightarrow 0\ n−p1​∑k=1n​(dk​−d)→0  almost surely with p∈(1,2)p\in(1,2)p∈(1,2), are developed for products dk=∏r=1sxk(r)d_k=\prod_{r=1}^s x_k^{(r)}dk​=∏r=1s​xk(r)​, where the xk(r)=∑l=−∞∞ck−l(r)ξl(r)x_k^{(r)} = \sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}\xi_l^{(r)}xk(r)​=∑l=−∞∞​ck−l(r)​ξl(r)​ are two-sided linear process with coefficients {cl(r)}l∈Z\{c_l^{(r)}\}_{l\in \mathbb{Z}}{cl(r)​}l∈Z​ and i.i.d. zero-mean innovations {ξl(r)}l∈Z\{\xi_l^{(r)}\}_{l\in \mathbb{Z}}{ξl(r)​}l∈Z​. The decay of the coefficients cl(r)c_l^{(r)}cl(r)​ as ∣l∣→∞|l|\to\infty∣l∣→∞, can be slow enough for {xk(r)}\{x_k^{(r)}\}{xk(r)​} to have long memory while {dk}\{d_k\}{dk​} can have heavy tails. The long-range dependence and heavy tails for {dk}\{d_k\}{dk​} 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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