{"paper":{"title":"Stable size-biasing and the positive scale-mixture order of generalized Gaussian laws","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Domingos S. P. Salazar","submitted_at":"2026-06-16T20:04:47Z","abstract_excerpt":"Let $X_r\\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\\exp(-|x|^r/2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_p\\stackrel{d}{=}VX_q$ if and only if $p\\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1/p$, $b=1/q$, and $\\alpha=b/a=p/q$. If $S_\\alpha$ is a positive $\\alpha$-stable random variable with Laplace transform $\\mathbb{E}\\exp(-uS_\\alpha)=\\exp(-u^\\alpha)$, set $W_0=S_\\alpha^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18458","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.18458/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}