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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1411.1312","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2014-11-05T16:25:59Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"584037dd3f0cecd9a74397eb3e49cea4f6ec591ef4ac94ff11d35170c281c7d5","abstract_canon_sha256":"237c393c77b336a2539bfc645f50bb273c155e3d2a8a42ac7e0447833abe8500"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:30.455331Z","signature_b64":"bbMfDMiAQleb7q7/vJU+4Ju5AlAIRZXo8asGVesEDUx5hdt1aj+N4m6WlRRQIcGvC8jzzCD1kkoAe3ovqXAbCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"808f2fb8a4491acd2cad34a7072c3ba75299944346bd7486f577481c65ac0ac9","last_reissued_at":"2026-05-18T02:32:30.454942Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:30.454942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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