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We observe that $Z_N=X_1+ \\ldots + X_N$ for independent and bounded random variables $X_j$'s, so $Z_N$ has finite mean ${\\bf E}Z_N$ and variance ${\\bf Var}Z_N$. We want to find the limit distribution of ${\\hat Z}_N=\\left(Z_N-{\\bf E}Z_N\\right)/{\\sqrt{{\\bf Var}Z_N}}$ as $N \\to \\infty.$ While in many cases the limit distribution is normal, the main results established in this paper are that ${\\hat Z}_N \\overset{d}{\\to} Z_{*},$ where $Z_{*}$ is a bound"},"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":"1901.04029","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-01-13T18:04:53Z","cross_cats_sorted":[],"title_canon_sha256":"06d82cc02a61bdfee4ca568e2339717d618eaec559c0c3b697349362727b2654","abstract_canon_sha256":"9679e194fe49365097fc1c51447a9c0a9651d0afa7cdfac87ee237b44a34778d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:26.468569Z","signature_b64":"MR2DXdNhyFDlLzaMSuCzJNI5q2IccwiNTZFCPJamB0dJ1Jv7l7mDT4ZJvjp7asMm2MuocpRUb+n4Kr7bS5TCAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cba6cd5b1ab84ce8511357e3bfd54ba4e7511099f8a3eb39363118de6bc613ea","last_reissued_at":"2026-05-17T23:56:26.467980Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:26.467980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-Conventional Limits of Random Sequences Related to Partitions of Integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"C. 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