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Given $N$ independent copies $(A_i,Z_i)$ of $(A,Z)$ we define a new random variable $\\hat Z = \\sum_{i=1}^N A_i Z_i$. Let $T$ be the corresponding transformation on the set of probability measures on $C$ i.e. $T$ maps the law of $Z$ to the law of $\\hat Z$. If the matrix $\\mathbb{E}[N] \\mathbb{E} [A]$ has dominant eigenvalue 1, we study existence and properties of fixed points of $T$ having finite"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1109.1845","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-08T20:59:05Z","cross_cats_sorted":[],"title_canon_sha256":"fd5fefdb12c6566b388b7c37b805308f5a26a5c5eec535a5f69cd8d72b322791","abstract_canon_sha256":"2be8407f8e7ed07122e8cb06b85dc24021054590f29e39de7c6788d34b2d4891"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:33.043723Z","signature_b64":"hZlEdIYRrLtwzJbl1Gqk3cAJDE26hyLq6B5fNCQrDihmc3PUtJqSLebZRqdA5G9gXlS7awFad2c+LddHt5MuAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"248beab718a172c44586522caf5f093200b25f622f195b7bad5a5ef4f3b8b338","last_reissued_at":"2026-05-18T02:56:33.043088Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:33.043088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On multidimensional Mandelbrot's cascades","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dariusz Buraczewski, Ewa Damek, Sebastian Mentemeier, Yves Guivarc'h","submitted_at":"2011-09-08T20:59:05Z","abstract_excerpt":"Let $Z$ be a random variable with values in a proper closed convex cone $C\\subset \\mathbb{R}^d$, $A$ a random endomorphism of $C$ and $N$ a random integer. 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