{"paper":{"title":"A Generalization of an Integral Arising in the Theory of Distance Correlation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Dominic Edelmann, Donald Richards, Johannes Dueck","submitted_at":"2014-11-05T16:25:59Z","abstract_excerpt":"We generalize an integral which arises in several areas in probability and statistics and which is at the core of the field of distance correlation, a concept developed by Sz\\'ekely, Rizzo and Bakirov (2007) to measure dependence between random variables. Let $m$ be a positive integer and let ${\\cos_m}(u)$, $u \\in \\mathbb{R}$, be the truncated Maclaurin expansion of ${\\cos}(u)$, where the expansion is truncated at the $m$th summand. For $t, x \\in \\mathbb{R}^d$, let $\\langle t,x\\rangle$ and $\\|x\\|$ denote the standard Euclidean inner product and norm, respectively. We establish the integral for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1312","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}