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In particular, we prove weak convergence of finite-dimensional distributions for the process $(X_n ([k_n u]))_{u>0}$, properly normalized and centered, as $n\\to\\infty$. The limit is a centered Gaussian process with covariance $(u,v)\\mapsto (u+v)^{-1}$. One-dimensional distributional convergence of $X_n(k_n)$, properly normalized and centered, was"},"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":"1806.10872","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-06-28T10:43:54Z","cross_cats_sorted":[],"title_canon_sha256":"a8d977a5aaf69d1a7f1036f774051b31741cc62214e87115d3158609c3603a63","abstract_canon_sha256":"92a9a80b1b77cd1fe7d582798abe232718f74166192618e5c771c1c6e757400e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:07.727673Z","signature_b64":"gQoSheOq7i6l0uyOBq0A/AS1RpEWNZ8tiS2qVy4tpCgHhQy/BCjP3H+S5QCL8GRNkO+m0+USUCIhF2TnQOYrCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7fb8c3cf0dada19a021c2d43159f0e1d97f4db2274c8023636ec9a4135deebc3","last_reissued_at":"2026-05-18T00:12:07.727001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:07.727001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak convergence of the number of vertices at intermediate levels of random recursive trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Iksanov, Zakhar Kabluchko","submitted_at":"2018-06-28T10:43:54Z","abstract_excerpt":"Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. 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