{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:52:18Z","timestamp":1760241138678,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,12,6]],"date-time":"2019-12-06T00:00:00Z","timestamp":1575590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation     X =  \u2211  k \u2265 1    \u03bb k   (  \u03be k 2  \u2212 1 )     , where      \u03be k  \u223c N  ( 0 , 1 )      are independent. We compute the L\u00e9vy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever      \u03bb k  \u2260 0     infinitely often. Our work generalises some known results presented in the literature.<\/jats:p>","DOI":"10.3390\/sym11121487","type":"journal-article","created":{"date-parts":[[2019,12,6]],"date-time":"2019-12-06T10:41:44Z","timestamp":1575628904000},"page":"1487","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on Distributions in the Second Chaos"],"prefix":"10.3390","volume":"11","author":[{"given":"Pauliina","family":"Ilmonen","sequence":"first","affiliation":[{"name":"Department of Mathematics and Systems Analysis, Aalto University School of Science, 00076 Aalto, Finland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3015-8664","authenticated-orcid":false,"given":"Lauri","family":"Viitasaari","sequence":"additional","affiliation":[{"name":"Department of Information and Service Management, Aalto University School of Business, 00076 Aalto, Finland"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1767","DOI":"10.1214\/aos\/1176347394","article-title":"The empirical process of some long-range dependent sequences with an application to U-statistics","volume":"17","author":"Dehling","year":"1989","journal-title":"Ann. 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