{"paper":{"title":"Precise tail index of fixed points of the two-sided smoothing transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ewa Damek, Gerold Alsmeyer, Sebastian Mentemeier","submitted_at":"2012-06-18T15:42:10Z","abstract_excerpt":"We consider real-valued random variables R satisfying the distributional equation\nR \\eqdist \\sum_{k=1}^{N}T_k R_k + Q,\nwhere R_1,R_2,... are iid copies of R and independent of T=(Q, (T_k)_{k \\ge 1}). N is the number of nonzero weights T_k and assumed to be a.s. finite. Its properties are governed by the function m(s) := \\E \\sum_{k=1}^N |T_k|^s . There are at most two values \\alpha < \\beta such that m(\\alpha)=m(\\beta)=1. We consider solutions R with finite moment of order s > \\alpha. We review results about existence and uniqueness. Assuming the existence of \\beta and an additional mild moment "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3970","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"}