{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:35:27Z","timestamp":1759970127291,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,7]],"date-time":"2025-01-07T00:00:00Z","timestamp":1736208000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["586789","12031005"],"award-info":[{"award-number":["586789","12031005"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Nature Science Foundation of China","doi-asserted-by":"publisher","award":["586789","12031005"],"award-info":[{"award-number":["586789","12031005"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shenzhen Outstanding Talents Training Fund","award":["586789","12031005"],"award-info":[{"award-number":["586789","12031005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we study self-normalized moderate deviations for degenerate U-statistics of order 2. Let {Xi,i\u22651} be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form h(x,y)=\u2211l=1\u221e\u03bblgl(x)gl(y), where \u03bbl&gt;0, Egl(X1)=0, and gl(X1) is in the domain of attraction of a normal law for all l\u22651. Under the condition \u2211l=1\u221e\u03bbl&lt;\u221e and some truncated conditions for {gl(X1):l\u22651}, we show that logP(\u22111\u2264i\u2260j\u2264nh(Xi,Xj)max1\u2264l&lt;\u221e\u03bblVn,l2\u2265xn2)\u223c\u2212xn22 for xn\u2192\u221e and xn=o(n), where Vn,l2=\u2211i=1ngl2(Xi). As application, a law of the iterated logarithm is also obtained.<\/jats:p>","DOI":"10.3390\/e27010041","type":"journal-article","created":{"date-parts":[[2025,1,7]],"date-time":"2025-01-07T05:06:34Z","timestamp":1736226394000},"page":"41","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Self-Normalized Moderate Deviations for Degenerate U-Statistics"],"prefix":"10.3390","volume":"27","author":[{"given":"Lin","family":"Ge","sequence":"first","affiliation":[{"name":"Division of Arts and Sciences, Mississippi State University at Meridian, Meridian, MS 39307, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9155-4636","authenticated-orcid":false,"given":"Hailin","family":"Sang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Mississippi, University, MS 38677, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qi-Man","family":"Shao","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1571","DOI":"10.1214\/aop\/1176991175","article-title":"Self-normalized laws of the iterated logarithm","volume":"17","author":"Griffin","year":"1989","journal-title":"Ann. 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