What is recorded here?
Hoeffding 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. Averages 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.
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Scientific scope
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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- Hoeffding: bounded-sum concentration and convex sampling comparison — original scientific article
- Hoeffding: bounded-sum concentration and convex sampling comparison — publisher deposit
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People and systems
Wassily Hoeffding
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English catalogue narrative
Hoeffding: bounded-sum concentration and convex sampling comparison
Hoeffding 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.
Averages 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.
Scientific scope
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.
People and systems
Wassily Hoeffding