{"paper":{"title":"Super-Gaussian directions of random vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Bo'az Klartag","submitted_at":"2015-12-10T15:12:16Z","abstract_excerpt":"We establish the following universality property in high dimensions: Let $X$ be a random vector with density in $\\mathbb{R}^n$. The density function can be arbitrary. We show that there exists a fixed unit vector $\\theta \\in \\mathbb{R}^n$ such that the random variable $Y = \\langle X, \\theta \\rangle$ satisfies $$ \\min \\left \\{ \\mathbb{P}( Y \\geq t M ), \\mathbb{P}(Y \\leq -tM) \\right \\} \\geq c e^{-C t^2} \\qquad \\qquad \\text{for all} \\ 0 \\leq t \\leq \\tilde{c} \\sqrt{n}, $$ where $M > 0$ is any median of $|Y|$, i.e., $\\min \\{ \\mathbb{P}( |Y| \\geq M), \\mathbb{P}( |Y| \\leq M ) \\} \\geq 1/2$. Here, $c, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03282","kind":"arxiv","version":2},"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"}