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In particular, a Gaussian approximation result is established with a nonasymptotic error bound, and the asymptotic null distribution of the sample generalized distance covariance is shown to be distributed as a linear combination of independently and identically distributed chi-squared random variables. To estimate the asymptotic null distribution practically, we propose a half-permutation procedure and provide the theoretical justification for its validity. The exact asymptotic distribution of the resampling distribution is derived under general marginal moment conditions, and the proposed procedure is shown to be asymptotically equivalent to the oracle procedure with known marginal distributions.<\/jats:p>","DOI":"10.1214\/25-aos2613","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:50:36Z","timestamp":1782395436000},"page":"1451-1473","source":"Crossref","is-referenced-by-count":1,"title":["Test of independence using generalized distance correlation"],"prefix":"10.1214","volume":"54","author":[{"given":"Jianqing","family":"Fan","sequence":"first","affiliation":[{"name":"Department of Operations Research and Financial Engineering, Princeton University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhipeng","family":"Lou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, San Diego"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Danna","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, San Diego"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"6","doi-asserted-by":"publisher","unstructured":"Esseen, C.-G.<\/small> (1945). 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