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        "description": "The concentration bounds apply to the specified independent bounded variables and separately treated sampling settings. Bounded ranges, means and dependence assumptions cannot be omitted. The bounds are distribution-free within those conditions, not a proof for arbitrary adaptive/dependent sequences. Later martingale formulations and the withheld Azuma record are not silently credited here.",
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      "statement": "# Hoeffding: bounded-sum concentration and convex sampling comparison\n\nHoeffding organized finite tail bounds for bounded independent sums, extended a range-only bound to unequal intervals, and transferred exponential bounds to explicitly represented dependent statistics and finite-population sampling.\n\nAverages can be reliable before a normal approximation is accurate. If each independent summand lies in a known finite interval, a convex majorant of its exponential moment yields a probability bound using the sum of squared interval widths. The paper also distinguishes sharper mean-sensitive bounds from variance-sensitive bounds for variables bounded only above. These are finite inequalities, not declarations that the bound equals the actual tail probability. The same article explicitly notes a martingale extension and uses an inherited maximal inequality. For U statistics, it averages sums built from disjoint observations; an arbitrary dependent sum has no such automatic guarantee. For a uniform sample without replacement from a finite population, it proves a continuous-convex comparison against sampling with replacement, allowing existing exponential bounds to transfer. The latter comparison reduces uncertainty without claiming that the resulting bound is optimal.\n\n## Scientific scope\n\nThe concentration bounds apply to the specified independent bounded variables and separately treated sampling settings. Bounded ranges, means and dependence assumptions cannot be omitted. The bounds are distribution-free within those conditions, not a proof for arbitrary adaptive/dependent sequences. Later martingale formulations and the withheld Azuma record are not silently credited here.\n\n## People and systems\n\nWassily Hoeffding\n\n\n\n## Original sources\n\n- [Hoeffding: bounded-sum concentration and convex sampling comparison — original scientific article](https://www.cs.rpi.edu/academics/courses/spring06/random/hoefding.pdf)\n- [Hoeffding: bounded-sum concentration and convex sampling comparison — publisher deposit](https://api.crossref.org/works/10.1080%2F01621459.1963.10500830)\n",
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