On highly skewed fractional log-stable noise sequences and their application

IF 1.2 4区 数学 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Harry Pavlopoulos, George Chronis
{"title":"On highly skewed fractional log-stable noise sequences and their application","authors":"Harry Pavlopoulos,&nbsp;George Chronis","doi":"10.1111/jtsa.12671","DOIUrl":null,"url":null,"abstract":"<p>Considering log-LFSN (log-linear fractional stable noise) sequences <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>Y</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>=</mo>\n <msup>\n <mrow>\n <mi>e</mi>\n </mrow>\n <mrow>\n <mi>δ</mi>\n <mo>·</mo>\n <msub>\n <mrow>\n <mi>X</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>+</mo>\n <mi>ε</mi>\n </mrow>\n </msup>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math>, driven by non-Gaussian one-sided LFSN <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>X</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math> with constant skewness intensity <math>\n <msub>\n <mrow>\n <mi>β</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>∈</mo>\n <mo>[</mo>\n <mo>−</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>]</mo></math>, for any <math>\n <mi>δ</mi>\n <mo>∈</mo>\n <mi>ℝ</mi>\n <mo>−</mo>\n <mo>{</mo>\n <mn>0</mn>\n <mo>}</mo></math> and <math>\n <mi>ε</mi>\n <mo>∈</mo>\n <mi>ℝ</mi></math>, we show that the auto-covariance function (ACVF) <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>γ</mi>\n </mrow>\n <mrow>\n <mi>Y</mi>\n </mrow>\n </msub>\n <mo>(</mo>\n <mi>h</mi>\n <mo>)</mo>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>h</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math> exists if and only if <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>X</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math> is persistent, with stability index <math>\n <mi>α</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mn>2</mn>\n <mo>)</mo></math>, Hurst exponent <math>\n <mi>H</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mi>α</mi>\n <mo>,</mo>\n <mn>1</mn>\n <mo>)</mo></math> and extreme skewness <math>\n <msub>\n <mrow>\n <mi>β</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>=</mo>\n <mo>−</mo>\n <mn>1</mn></math> (if <math>\n <mi>δ</mi>\n <mo>&gt;</mo>\n <mn>0</mn></math>) or <math>\n <msub>\n <mrow>\n <mi>β</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>=</mo>\n <mn>1</mn></math> (if <math>\n <mi>δ</mi>\n <mo>&lt;</mo>\n <mn>0</mn></math>). Within that range of existence, <math>\n <mn>1</mn>\n <mo>/</mo>\n <mn>2</mn>\n <mo>&lt;</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mi>α</mi>\n <mo>&lt;</mo>\n <mi>H</mi>\n <mo>&lt;</mo>\n <mn>1</mn></math> and <math>\n <mo>|</mo>\n <msub>\n <mrow>\n <mi>β</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>|</mo>\n <mo>=</mo>\n <mn>1</mn></math> in short, we calculate <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>γ</mi>\n </mrow>\n <mrow>\n <mi>Y</mi>\n </mrow>\n </msub>\n <mo>(</mo>\n <mi>h</mi>\n <mo>)</mo>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>h</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math> explicitly and establish persistence of <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>Y</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math> too, by showing asymptotic proportionality of <math>\n <msub>\n <mrow>\n <mi>γ</mi>\n </mrow>\n <mrow>\n <mi>Y</mi>\n </mrow>\n </msub>\n <mo>(</mo>\n <mi>h</mi>\n <mo>)</mo>\n <mo>≈</mo>\n <mo>|</mo>\n <mi>h</mi>\n <msup>\n <mrow>\n <mo>|</mo>\n </mrow>\n <mrow>\n <mi>α</mi>\n <mo>·</mo>\n <mo>(</mo>\n <mi>H</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n </msup></math>, as <math>\n <mi>h</mi>\n <mo>→</mo>\n <mi>∞</mi></math>. We discuss explicit links of <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>γ</mi>\n </mrow>\n <mrow>\n <mi>Y</mi>\n </mrow>\n </msub>\n <mo>(</mo>\n <mi>h</mi>\n <mo>)</mo>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>h</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math> to a generalized co-difference function of the driving one-sided LFSN <math>\n <msub>\n <mrow>\n <mo>{</mo>\n <msub>\n <mrow>\n <mi>X</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msub>\n <mo>}</mo>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </mrow>\n </msub></math>, and to the ACVF's of fractional Gaussian noise (FGN) and log-FGN. The results are numerically demonstrated via ensemble simulation of synthetic time series generated by the considered log-LFSN model fitted to time series of spatio-temporal accumulations of rain rate data.</p>","PeriodicalId":49973,"journal":{"name":"Journal of Time Series Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2022-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/jtsa.12671","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Time Series Analysis","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/jtsa.12671","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0

Abstract

Considering log-LFSN (log-linear fractional stable noise) sequences { Y n = e δ · X n + ε } n , driven by non-Gaussian one-sided LFSN { X n } n with constant skewness intensity β 0 [ 1 , 1 ], for any δ { 0 } and ε , we show that the auto-covariance function (ACVF) { γ Y ( h ) } h exists if and only if { X n } n is persistent, with stability index α ( 1 , 2 ), Hurst exponent H ( 1 / α , 1 ) and extreme skewness β 0 = 1 (if δ > 0) or β 0 = 1 (if δ < 0). Within that range of existence, 1 / 2 < 1 / α < H < 1 and | β 0 | = 1 in short, we calculate { γ Y ( h ) } h explicitly and establish persistence of { Y n } n too, by showing asymptotic proportionality of γ Y ( h ) | h | α · ( H 1 ) , as h . We discuss explicit links of { γ Y ( h ) } h to a generalized co-difference function of the driving one-sided LFSN { X n } n , and to the ACVF's of fractional Gaussian noise (FGN) and log-FGN. The results are numerically demonstrated via ensemble simulation of synthetic time series generated by the considered log-LFSN model fitted to time series of spatio-temporal accumulations of rain rate data.

Abstract Image

关于高偏斜分数对数稳定噪声序列及其应用
考虑对数线性分数稳定噪声序列{Yn=eδ·Xn+ε}n∈ℤ , 由非高斯单侧LFSN驱动{Xn}n∈ℤ 具有恒定偏度强度β0∈[-1,1],对于任何δ∈ℝ−{0}和ε∈ℝ , 我们证明了自协方差函数(ACVF){γY(h)}h∈ℤ 存在当且仅当{Xn}n∈ℤ 是持久的,稳定指数α∈(1,2),赫斯特指数H∈(1/α,1)和极偏β0=−1(如果δ>0)或β0=1(如果Δ<0)。在该存在范围内,1/2<1/α
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来源期刊
Journal of Time Series Analysis
Journal of Time Series Analysis 数学-数学跨学科应用
CiteScore
2.00
自引率
0.00%
发文量
39
审稿时长
6-12 weeks
期刊介绍: During the last 30 years Time Series Analysis has become one of the most important and widely used branches of Mathematical Statistics. Its fields of application range from neurophysiology to astrophysics and it covers such well-known areas as economic forecasting, study of biological data, control systems, signal processing and communications and vibrations engineering. The Journal of Time Series Analysis started in 1980, has since become the leading journal in its field, publishing papers on both fundamental theory and applications, as well as review papers dealing with recent advances in major areas of the subject and short communications on theoretical developments. The editorial board consists of many of the world''s leading experts in Time Series Analysis.
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