{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:P64MHTYNVWQZUAQ4FVBRLHYODW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"92a9a80b1b77cd1fe7d582798abe232718f74166192618e5c771c1c6e757400e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-06-28T10:43:54Z","title_canon_sha256":"a8d977a5aaf69d1a7f1036f774051b31741cc62214e87115d3158609c3603a63"},"schema_version":"1.0","source":{"id":"1806.10872","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.10872","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"arxiv_version","alias_value":"1806.10872v1","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.10872","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"pith_short_12","alias_value":"P64MHTYNVWQZ","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"P64MHTYNVWQZUAQ4","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"P64MHTYN","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:6a8dcbfea4520b334187dd22f90cb62ecd18709347f129061ed9cb2fddf0ab8a","target":"graph","created_at":"2026-05-18T00:12:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We are interested in the asymptotic behavior of $X_n(k)$ for intermediate levels $k=k_n$ satisfying $k_n\\to\\infty$ and $k_n=o(\\log n)$ as $n\\to\\infty$. 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","authors_text":"Alexander Iksanov, Zakhar Kabluchko","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-06-28T10:43:54Z","title":"Weak convergence of the number of vertices at intermediate levels of random recursive trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10872","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2c33d8f5fb4896a40d205fe7ba2a0515cbb3352e29c651f30bd2e71e576ded8a","target":"record","created_at":"2026-05-18T00:12:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"92a9a80b1b77cd1fe7d582798abe232718f74166192618e5c771c1c6e757400e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-06-28T10:43:54Z","title_canon_sha256":"a8d977a5aaf69d1a7f1036f774051b31741cc62214e87115d3158609c3603a63"},"schema_version":"1.0","source":{"id":"1806.10872","kind":"arxiv","version":1}},"canonical_sha256":"7fb8c3cf0dada19a021c2d43159f0e1d97f4db2274c8023636ec9a4135deebc3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7fb8c3cf0dada19a021c2d43159f0e1d97f4db2274c8023636ec9a4135deebc3","first_computed_at":"2026-05-18T00:12:07.727001Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:07.727001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gQoSheOq7i6l0uyOBq0A/AS1RpEWNZ8tiS2qVy4tpCgHhQy/BCjP3H+S5QCL8GRNkO+m0+USUCIhF2TnQOYrCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:07.727673Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.10872","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c33d8f5fb4896a40d205fe7ba2a0515cbb3352e29c651f30bd2e71e576ded8a","sha256:6a8dcbfea4520b334187dd22f90cb62ecd18709347f129061ed9cb2fddf0ab8a"],"state_sha256":"3bad0e502694d6a70a2f8a03cf2f1d9c9c33b9d87e6a7e80421056b67a7d288f"}