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The large deviation principle (LDP) for such projections as $n\\rightarrow\\infty$ is given by the classical Cram\\'er's theorem. We prove an LDP for the sequence of normalized scalar projections of $X^{(n)}$ in the direction of a generic unit vector $\\theta^{(n)} \\in \\mathbb{S}^{n-1}$"},"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":"1508.04402","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-18T18:19:18Z","cross_cats_sorted":[],"title_canon_sha256":"7dee13216ceee2af42fad11a910066e88aaf8c1953dad58cabb89b9cce8ef58b","abstract_canon_sha256":"0ccb2894c55dd93d46179443551d5d154aed20c027e34bb0d87182084047d31f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:57.841706Z","signature_b64":"+HJiTgwOj8r8TXyKOU79LXnpJ34J7scxY565yoKf6xjBydNU/dr3Mk/5kCkS11tCUX4rlBDAVSZJmCBvhoadDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba70eefbd6e6f5b2782c0dfe6b7ae879b1c2456b78aac5c5127c10a898c90275","last_reissued_at":"2026-05-18T01:30:57.841080Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:57.841080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cram\\'er's theorem is atypical","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kavita Ramanan, Nina Gantert, Steven Soojin Kim","submitted_at":"2015-08-18T18:19:18Z","abstract_excerpt":"The empirical mean of $n$ independent and identically distributed (i.i.d.) random variables $(X_1,\\dots,X_n)$ can be viewed as a suitably normalized scalar projection of the $n$-dimensional random vector $X^{(n)}\\doteq(X_1,\\dots,X_n)$ in the direction of the unit vector $n^{-1/2}(1,1,\\dots,1) \\in \\mathbb{S}^{n-1}$. The large deviation principle (LDP) for such projections as $n\\rightarrow\\infty$ is given by the classical Cram\\'er's theorem. 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