{"paper":{"title":"Fixed Points of the Multivariate Smoothing Transform: The Critical Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konrad Kolesko, Sebastian Mentemeier","submitted_at":"2014-09-25T11:46:29Z","abstract_excerpt":"Given a sequence $(T_1, T_2, ...)$ of random $d \\times d$ matrices with nonnegative entries, suppose there is a random vector $X$ with nonnegative entries, such that $ \\sum_{i \\ge 1} T_i X_i $ has the same law as $X$, where $(X_1, X_2, ...)$ are i.i.d. copies of $X$, independent of $(T_1, T_2, ...)$. Then (the law of) $X$ is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case $d=1$, a function $m$ is introduced, such that the existence of $\\alpha \\in (0,1]$ with $m(\\alpha)=1$ and $m'(\\alpha) \\le 0$ guarantees the existence of nontrivia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7220","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":""},"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"}