{"title":"On highly skewed fractional log-stable noise sequences and their application","authors":"Harry Pavlopoulos, 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>></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><</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><</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mi>α</mi>\n <mo><</mo>\n <mi>H</mi>\n <mo><</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 , driven by non-Gaussian one-sided LFSN with constant skewness intensity , for any and , we show that the auto-covariance function (ACVF) exists if and only if is persistent, with stability index , Hurst exponent and extreme skewness (if ) or (if ). Within that range of existence, and in short, we calculate explicitly and establish persistence of too, by showing asymptotic proportionality of , as . We discuss explicit links of to a generalized co-difference function of the driving one-sided LFSN , 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.
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.