{"title":"Fractional Skellam Process of Order k","authors":"","doi":"10.1007/s10959-024-01314-8","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We introduce and study a fractional version of the Skellam process of order <em>k</em> by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order <em>k</em> (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of FSPoK which are utilized to establish its long-range dependence property. Later, we consider two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes. </p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"101 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Theoretical Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10959-024-01314-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce and study a fractional version of the Skellam process of order k by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order k (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of FSPoK which are utilized to establish its long-range dependence property. Later, we consider two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes.
期刊介绍：
Journal of Theoretical Probability publishes high-quality, original papers in all areas of probability theory, including probability on semigroups, groups, vector spaces, other abstract structures, and random matrices. This multidisciplinary quarterly provides mathematicians and researchers in physics, engineering, statistics, financial mathematics, and computer science with a peer-reviewed forum for the exchange of vital ideas in the field of theoretical probability.
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