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Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \\[Z_N(t)=\\sum_{i=1}^N\\mathrm{e}^{\\xi_i(s_N+t)}\\] as $N\\to\\infty$, where $s_N$ is a non-negative sequence converging to $+\\infty$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\\liminf_{N\\to\\infty}\\log N/s_N>\\lambda_2$ for some critical value $\\lambda_2>0$, then the limit is an Ornstein--U"},"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":"0911.4139","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-11-21T16:27:31Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"1ba575fab75453c89afc58bee68e464de88d897b8937d6425ad853bab42854ef","abstract_canon_sha256":"bfbd706d23e0fd505d0d24e454a0e613bbd62f0c12dc28ce3920bef935991108"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:22.570028Z","signature_b64":"xWRfvPxMQ2YWET+RaluTu/NM3R7jpMK5MvmzUswpdNpKxx8V4ji36eQg1RwsEDRcJwiGCeFuUzpCSdoLafYmBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7262f715edece355adae8ea842e3b1c8848c4fafb979b679a1515a5b8b02c97e","last_reissued_at":"2026-05-18T04:18:22.569480Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:22.569480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Functional limit theorems for sums of independent geometric L\\'{e}vy processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Zakhar Kabluchko","submitted_at":"2009-11-21T16:27:31Z","abstract_excerpt":"Let $\\xi_i$, $i\\in \\mathbb {N}$, be independent copies of a L\\'{e}vy process $\\{\\xi(t),t\\geq0\\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \\[Z_N(t)=\\sum_{i=1}^N\\mathrm{e}^{\\xi_i(s_N+t)}\\] as $N\\to\\infty$, where $s_N$ is a non-negative sequence converging to $+\\infty$. The limiting process depends heavily on the growth rate of the sequence $s_N$. 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