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    "title": "Azuma: weighted dependent sums and asymptotic fluctuation control",
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    "area": "Mathematics and statistics",
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        "description": "The 1967 weighted-sum and asymptotic contribution. The exponential estimates do not erase the martingale extension already noted in the 1963 Hoeffding record. Theorem2 uses D_n^2=sum a_i^2, a_n^2/D_n^2->0 and D_n^2->infinity; Theorem3 uses positive increasing weights with a_n/A_n=o(1/log log A_n). These are upper/convergence conclusions, not an unconditional sharp two-sided LIL. Tsuchikura supplied a separating weight example.",
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      "statement": "# Azuma: weighted dependent sums and asymptotic fluctuation control\n\nIndependent errors are easier to average than errors that depend on the past. Azuma studies sequences whose next increment has zero conditional mean, together with boundedness or conditional exponential-moment assumptions. These conditions control weighted sums and how their fluctuations grow. The paper gives convergence and iterated-logarithm upper bounds under explicit restrictions on the weights. This adds a route for studying dependent data without pretending that every dependent sequence behaves like independent noise.\n\n## Scientific scope\n\nThe 1967 weighted-sum and asymptotic contribution. The exponential estimates do not erase the martingale extension already noted in the 1963 Hoeffding record. Theorem2 uses D_n^2=sum a_i^2, a_n^2/D_n^2->0 and D_n^2->infinity; Theorem3 uses positive increasing weights with a_n/A_n=o(1/log log A_n). These are upper/convergence conclusions, not an unconditional sharp two-sided LIL. Tsuchikura supplied a separating weight example.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nKazuoki Azuma\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Azuma: exponential control of bounded dependent weighted sums — original scientific article](https://www.jstage.jst.go.jp/article/tmj1949/19/3/19_3_357/_pdf/-char/en)\n- [Azuma: exponential control of bounded dependent weighted sums — publisher deposit](https://api.crossref.org/works/10.2748%2Ftmj%2F1178243286)\n",
      "summary": "Independent errors are easier to average than errors that depend on the past. Azuma studies sequences whose next increment has zero conditional mean, together with boundedness or conditional exponential-moment assumptions. These conditions control weighted sums and how their fluctuations grow. The paper gives convergence and iterated-logarithm upper bounds under explicit restrictions on the weights. This adds a route for studying dependent data without pretending that every dependent sequence behaves like independent noise.",
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    "what_changed": "Independent errors are easier to average than errors that depend on the past. Azuma studies sequences whose next increment has zero conditional mean, together with boundedness or conditional exponential-moment assumptions. These conditions control weighted sums and how their fluctuations grow. The paper gives convergence and iterated-logarithm upper bounds under explicit restrictions on the weights. This adds a route for studying dependent data without pretending that every dependent sequence behaves like independent noise.",
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