{"paper":{"title":"Order statistics of vectors with dependent coordinates, and the Karhunen-Lo\\`eve basis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander E. Litvak, Konstantin Tikhomirov","submitted_at":"2016-09-07T19:27:18Z","abstract_excerpt":"Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not identically distributed coordinates and let $T$ be an orthogonal trasformation of $\\mathbb R^n$. We show that the random vector $Y=T(X)$ satisfies $$\\mathbb E\\sum\\limits_{j=1}^k j\\mbox{-}\\min_{i\\leq n}{X_{i}}^2 \\leq C\\mathbb E\\sum\\limits_{j=1}^k j\\mbox{-}\\min_{i\\leq n}{Y_{i}}^2$$ for all $k<n$, where \"$j\\mbox{-}\\min$\" denotes the $j$-th smallest component of corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02126","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"}