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To develop these approximations, we also generalize an important probability inequality of Rosenthal (1970,<jats:italic>Israel Journal of Mathematics<\/jats:italic>8, 273\u2013303; 1972,<jats:italic>Sixth Berkeley Symposium on Mathematical Statistics and Probability<\/jats:italic>, vol. 2, pp. 149\u2013167. University of California Press) to the case of Hilbert-space valued random variables.<\/jats:p>","DOI":"10.1017\/s0266466614000322","type":"journal-article","created":{"date-parts":[[2014,8,19]],"date-time":"2014-08-19T13:09:46Z","timestamp":1408453786000},"page":"337-361","source":"Crossref","is-referenced-by-count":16,"title":["THE INTEGRATED MEAN SQUARED ERROR OF SERIES REGRESSION AND A ROSENTHAL HILBERT-SPACE INEQUALITY"],"prefix":"10.1017","volume":"31","author":[{"given":"Bruce E.","family":"Hansen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,8,19]]},"reference":[{"key":"S0266466614000322_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0537-1"},{"key":"S0266466614000322_ref1","doi-asserted-by":"publisher","DOI":"10.2307\/2938259"},{"key":"S0266466614000322_ref20","volume-title":"Nonparametric Econometrics: Theory and Practice","author":"Li","year":"2006"},{"key":"S0266466614000322_ref26","first-page":"149","volume-title":"Sixth Berkeley Symposium on Mathematical Statistics and Probability","volume":"2","author":"Rosenthal","year":"1972"},{"key":"S0266466614000322_ref19","doi-asserted-by":"publisher","DOI":"10.1214\/lnms\/1215465643"},{"key":"S0266466614000322_ref21","doi-asserted-by":"crossref","first-page":"60","DOI":"10.4064\/fm-29-1-60-90","article-title":"Sur les foncions independantes","volume":"28","author":"Marcinkiewicz","year":"1937","journal-title":"Fund. 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