{"title":"Limit theorems for linear processes with tapered innovations and filters","authors":"","doi":"10.1007/s10986-024-09619-1","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We consider the partial-sum process <span> <span>\\({\\sum }_{k=1}^{\\left[nt\\right]}{X}_{k}^{\\left(n\\right)},\\)</span> </span> where <span> <span>\\(\\left\\{{X}_{k}^{\\left(n\\right)}={\\sum }_{j=0}^{\\infty }{\\alpha }_{j}^{\\left(n\\right)}{\\xi }_{k-j}\\left(b\\left(n\\right)\\right), k\\in {\\mathbb{Z}}\\right\\},\\)</span> </span> <em>n</em> ≥ 1, is a series of linear processes with tapered filter <span> <span>\\({\\alpha }_{j}^{\\left(n\\right)}={\\alpha }_{j} {1}_{\\left\\{0\\le j\\le\\lambda\\left(n\\right)\\right\\}}\\)</span> </span> and heavy-tailed tapered innovations <em>ξ</em><sub><em>j</em></sub>(<em>b</em>(<em>n</em>)), <em>j ∈</em> Z. Both tapering parameters <em>b</em>(<em>n</em>) and <em>⋋</em> (<em>n</em>) grow to <em>∞</em> as <em>n→∞</em>. The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter <em>a</em><sub><em>i</em></sub>, <em>i</em> ≥ 0, and nontapered innovations. We consider the cases where <em>b</em>(<em>n</em>) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of <em>⋋</em>(<em>n</em>) (strong, weak, and moderate tapering).</p>","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lithuanian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10986-024-09619-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the partial-sum process \({\sum }_{k=1}^{\left[nt\right]}{X}_{k}^{\left(n\right)},\) where \(\left\{{X}_{k}^{\left(n\right)}={\sum }_{j=0}^{\infty }{\alpha }_{j}^{\left(n\right)}{\xi }_{k-j}\left(b\left(n\right)\right), k\in {\mathbb{Z}}\right\},\)n ≥ 1, is a series of linear processes with tapered filter \({\alpha }_{j}^{\left(n\right)}={\alpha }_{j} {1}_{\left\{0\le j\le\lambda\left(n\right)\right\}}\) and heavy-tailed tapered innovations ξj(b(n)), j ∈ Z. Both tapering parameters b(n) and ⋋ (n) grow to ∞ as n→∞. The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter ai, i ≥ 0, and nontapered innovations. We consider the cases where b(n) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of ⋋(n) (strong, weak, and moderate tapering).
期刊介绍：
The Lithuanian Mathematical Journal publishes high-quality original papers mainly in pure mathematics. This multidisciplinary quarterly provides mathematicians and researchers in other areas of science with a peer-reviewed forum for the exchange of vital ideas in the field of mathematics.
The scope of the journal includes but is not limited to:
Probability theory and statistics;
Differential equations (theory and numerical methods);
Number theory;
Financial and actuarial mathematics, econometrics.
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