{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,3]],"date-time":"2026-10-03T18:29:43Z","timestamp":1791052183999,"version":"4.1.0"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,10,22]],"date-time":"2020-10-22T00:00:00Z","timestamp":1603324800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Council of scientific and industrial research India","award":["09\/1005(0021)2018-EMR-I"],"award-info":[{"award-number":["09\/1005(0021)2018-EMR-I"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Skellam process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, L\u00e9vy measures, governing difference-differential equations of the introduced processes. Our results generalize the Skellam process and running average of Poisson process in several directions.<\/jats:p>","DOI":"10.3390\/e22111193","type":"journal-article","created":{"date-parts":[[2020,10,22]],"date-time":"2020-10-22T10:27:58Z","timestamp":1603362478000},"page":"1193","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Skellam Type Processes of Order k and Beyond"],"prefix":"10.3390","volume":"22","author":[{"given":"Neha","family":"Gupta","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arun","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1932-4091","authenticated-orcid":false,"given":"Nikolai","family":"Leonenko","sequence":"additional","affiliation":[{"name":"Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"296","DOI":"10.2307\/2981372","article-title":"The frequency distribution of the difference between two Poisson variables belonging to different populations","volume":"109","author":"Skellam","year":"1946","journal-title":"J. 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