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Let $\\{\\mathbf Y_n\\}$ be the multi-dimensional Mandelbrot's martingale defined as sums of products of random matrixes indexed by nodes of "},"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":"1405.2681","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-05-12T09:15:15Z","cross_cats_sorted":[],"title_canon_sha256":"2ee6ab6df7f347d171939665c2be49e57c56b1f2583ee858a13348fff85dc6b1","abstract_canon_sha256":"108bf893147f7614593a8edef1ff06acd1c6e9c0f58281574b61cb9cd8e34ab8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:04.976129Z","signature_b64":"VmZPwRVKDTSeLPFt0MwKO7B4nxYtr1VExKawskPW09vW8ReOYRcO+iePLZZX0yLNr1lMCWTUsEC85FbpJINEDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da87b1f25f8b89a6be943fc5c2cb07ccd5457c87e58a60061fc15faa5c7e6275","last_reissued_at":"2026-05-18T02:52:04.975657Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:04.975657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Moments for multi-dimensional Mandelbrot's cascades","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chunmao Huang","submitted_at":"2014-05-12T09:15:15Z","abstract_excerpt":"We consider the distributional equation $\\textbf{Z}\\stackrel{d}{=}\\sum_{k=1}^N\\textbf{A}_k\\textbf{Z}(k) $, where $N$ is a random variable taking value in $\\mathbb N_0=\\{0,1,\\cdots\\}$, $\\textbf{A}_1,\\textbf{A}_2,\\cdots$ are $p\\times p$ non-negative random matrix, and $\\textbf{Z},\\textbf{Z}(1),\\textbf{Z}(2),\\cdots$ are $i.i.d$ random vectors in in $\\mathbb{R}_+^p$ with $\\mathbb{R}_+=[0,\\infty)$, which are independent of $(N,\\textbf{A}_1,\\textbf{A}_2,\\cdots)$. 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