Sharp small-deviation inequalities for sums of independent nonnegative random variables

Abstract: Let $(X_1,\ldots,X_n)$ be independent nonnegative random variables with $\mathbb{E} X_i\le1$, and write $S=\sum_iX_i$. For $δ>0$, we prove that [ \mathbb{P}\left(S<\mathbb{E} S+δ\right)\ge b_{n,δ}, ] where $b_{n,δ}=δ(n/(n+δ))^n$ for $0<δ<1$ and $b_{n,δ}=(1-1/(n+δ))^n$ for $δ\ge1$. The bound is sharp for every $n$ and $δ\ge 1$. In particular, since $b_{n,δ} \ge e^{-1}$ for $δ\ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $δ\ge 1$. The proof is found by ChatGPT 5.6 Pro. It combines the exact Dirichlet calibration theorem of Vlassis and Thomas [Vlassis and Thomas, 2026], which resolves Gaffke's conjecture in statistics, with results in convex geometry including Grünbaum's centroid theorem [Grünbaum, 1960] and its generalization by Letwin and Yaskin [Letwin and Yaskin, 2024].

• The paper establishes a sharp, dimension-dependent lower bound on the small-deviation probability for sums of independent nonnegative random variables.
• It employs advanced techniques including Dirichlet calibration and convex geometric tools, notably variants of Grünbaum’s centroid theorem, to derive explicit optimal constants.
• It resolves Feige’s conjecture for δ ≥ 1 through rigorous analysis and machine-checked proofs, with implications for probabilistic combinatorics and randomized algorithms.

Sharp Small-Deviation Inequalities for Sums of Independent Nonnegative Random Variables

Overview and Main Results

This work by Fu et al. ["Sharp small-deviation inequalities for sums of independent nonnegative random variables" (2607.23980)] provides a sharp, dimension-dependent lower bound on the small-deviation probability $\Pr(S < S + \delta)$ for $S = \sum_{i=1}^n X_i$ where $X_1, ..., X_n$ are independent nonnegative r.v.’s with $X_i \leq 1$. The authors resolve the Feige conjecture in the affirmative for $\delta \geq 1$, establish explicit optimal constants, and connect probabilistic inequalities with convex geometric tools, specifically variants of Grünbaum’s centroid theorem.

The main theorem asserts that for all $n \geq 1$ and $\delta > 0$,

$\Pr(S < S + \delta) \geq b_{n,\delta} := \begin{cases} \delta\left(\frac{n}{n+\delta}\right)^n, & 0 < \delta < 1, \[10pt] \left(1 - \frac{1}{n+\delta}\right)^n, & \delta \geq 1. \end{cases}$

This bound is sharp for every $n$ and $\delta \geq 1$. For $S = \sum_{i=1}^n X_i$0, $S = \sum_{i=1}^n X_i$1 for all $S = \sum_{i=1}^n X_i$2, proving the Feige conjecture for that regime. For $S = \sum_{i=1}^n X_i$3, the authors obtain an explicit bound $S = \sum_{i=1}^n X_i$4, which is weaker than Feige's conjecture but strictly universal.

Context, Conjectures, and Prior Work

The problem originates from efforts to generalize Markov- and Chebyshev-type tail bounds to settings that only enforce upper bounds and independence on nonnegative r.v.’s. Samuels initiated probabilistic analysis in this regime, and Feige [Feige 2004] formulated the sharpest possible universal small-deviation lower bound, predicting $S = \sum_{i=1}^n X_i$5 for all $S = \sum_{i=1}^n X_i$6. While previous works established lower bounds (initially $S = \sum_{i=1}^n X_i$7 and progressively up to $S = \sum_{i=1}^n X_i$8 for the unit-slack case), the sharp constant $S = \sum_{i=1}^n X_i$9 was elusive outside of additional regularity assumptions (e.g., i.i.d., log-concavity) [Guo et al. 2020, Alqasem et al. 2024, Egozcue and Fuentes Garcia 2025]. This work eliminates such constraints for $X_1, ..., X_n$0.

Technical Contributions

Exact Small-Deviation Bound

The authors fully resolve the sharpness of

$X_1, ..., X_n$1

for all $X_1, ..., X_n$2, demonstrating that the bound is tight via explicit extremal examples: i.i.d. two-point distributions with mass $X_1, ..., X_n$3 at $X_1, ..., X_n$4 and $X_1, ..., X_n$5 otherwise. For $X_1, ..., X_n$6, the bound

$X_1, ..., X_n$7

is established but does not match Feige's conjectured $X_1, ..., X_n$8 (for small $X_1, ..., X_n$9).

Proof Techniques

The proof synthesizes recent advances:

• Dirichlet Calibration: Leverages the Dirichlet calibration theorem of Vlassis and Thomas (Vlassis et al., 9 Jul 2026), which solves the Gaffke conjecture for distribution-free, bounded, nonnegative r.v.’s. This result provides the key probabilistic inequality.
• Convex Geometry and Grünbaum Inequalities: Applies Grünbaum’s centroid theorem [Grünbaum 1960] and its generalization by Letwin and Yaskin (Letwin et al., 2024), which control the volume of simplex halfspaces not passing through the centroid.
• Sharpness Construction: By examining two-point extremal distributions, the authors show optimality for all $X_i \leq 1$0.

The connection between the probability simplex (underlying the Dirichlet distribution) and the sums of bounded r.v.’s is pivotal, permitting the translation of geometric cap bounds into the sharp analytic inequality.

Formalization

The main theorem, and all geometric and probabilistic components, are formalized in Lean, providing an end-to-end machine-checked proof and a public codebase, pushing the rigor frontier for such inequalities.

Numerical Strength and Notable Claims

The work establishes that for all $X_i \leq 1$1 and $X_i \leq 1$2, the lower bound converges to $X_i \leq 1$3:

$X_i \leq 1$4

It further proves that this value is the sharp constant independent of $X_i \leq 1$5 at unit slack ($X_i \leq 1$6). For $X_i \leq 1$7,

$X_i \leq 1$8

so the result is strictly universal, although not optimal for small $X_i \leq 1$9.

A notable claim is the full resolution of Feige’s conjecture for $\delta \geq 1$0 for arbitrary independent nonnegative r.v.'s with $\delta \geq 1$1, without any further assumptions.

Implications and Future Directions

The resolution of Feige’s conjecture (for $\delta \geq 1$2) settles a core open problem in probabilistic combinatorics and the theory of sharp inequalities for independent r.v.'s, with immediate implications for randomized algorithms, tail estimates in bandits and statistics, and distribution-free confidence intervals.

Beyond direct applications, the embedding of probabilistic questions into convex geometry (utilizing simplex centroids and hyperplane sections) suggests a broad scope for transferring geometric inequalities into probabilistic bounds, potentially informing further advances for sums of dependent r.v.'s, concentration of measure, and stochastic optimization.

Longer-term, the modular proof strategy—coupling quantitative probabilistic and geometric statements—could stimulate sharper dimension-free bounds in various settings and inform the admissibility or asymptotic efficiency of new interval constructions, as recent work has already begun to examine [Ming et al. (Ming et al., 21 Jul 2026)].

Conclusion

Fu et al. establish sharp small-deviation inequalities for independent nonnegative bounded r.v.'s and rigorously resolve Feige’s conjecture for $\delta \geq 1$3 with optimal constants. The proof leverages Dirichlet calibration and convex geometry, illustrating the deep connections between probabilistic inequalities and geometric measure theory. The work both closes a major question and provides new technical tools and perspectives for future investigation in high-dimensional probability and applied statistics.