Feige's Conjecture in Probability

• Feige's Conjecture is a universal small-deviation inequality for sums of independent non-negative random variables with means at most one.
• The proof employs a reduction to two-point distributions to achieve tight, non-asymptotic lower bounds, even for skewed or heavy-tailed variables.
• This result has broad implications in finance, graph theory, and combinatorics by providing robust guarantees under minimal distributional assumptions.

Feige's Conjecture, in its probabilistic form, is a small-deviation inequality for sums of independent non-negative random variables. For arbitrary non-negative independent random variables $X_1,\dots,X_n$ with expectations $\mu_i \leq 1$, total mean $\mu=\sum_{i=1}^n \mu_i$, and any $\delta>0$, it asserts

$$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$

A 2025 paper gives a general proof for all such variables, all $n$, and all $\delta>0$, and presents the result as tight and fully non-asymptotic (Dürr, 10 Aug 2025).

1. Statement and probabilistic meaning

The conjecture concerns lower bounds on probability mass strictly below the level “expectation plus $\delta$.” The assumptions are minimal: independence, non-negativity, and the coordinatewise mean constraint $\mu_i\leq 1$. No variance bound, moment-growth condition, bounded-support assumption, or identical-distribution hypothesis is required (Dürr, 10 Aug 2025).

In this formulation, the lower bound is universal: $$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}, \qquad e^{-1}\approx 0.3679.$$ The result is described as a robust anti-concentration statement: even for highly skewed or heavy-tailed non-negative independent variables, one retains a guaranteed amount of mass below $\mu_i \leq 1$0 (Dürr, 10 Aug 2025).

A related small-deviation formulation appears in earlier work on the conjecture, where the problem is phrased in terms of independent mean-zero variables satisfying a one-sided lower bound and asks for a best universal constant $\mu_i \leq 1$1 (Guo et al., 2020). The 2025 proof resolves the non-negative formulation stated above in full generality (Dürr, 10 Aug 2025).

2. Extremal structure and proof strategy

The 2025 proof is organized around a reduction to extremal distribution classes. For a single non-negative random variable $\mu_i \leq 1$2 with mean $\mu_i \leq 1$3, the paper states that, for estimating $\mu_i \leq 1$4, it suffices to consider two-point distributions with one support point $\mu_i \leq 1$5 and the other $\mu_i \leq 1$6, preserving the mean via

$\mu_i \leq 1$7

This reduction is justified by a one-step algorithm that constructs a two-support random variable $\mu_i \leq 1$8 with the same mean while preserving the probability mass below the threshold (Dürr, 10 Aug 2025).

Within that reduced class, the minimizer is obtained by taking

$\mu_i \leq 1$9

Solving the mean constraint yields

$\mu=\sum_{i=1}^n \mu_i$0

and therefore

$\mu=\sum_{i=1}^n \mu_i$1

For one variable, this lower bound is stated to be tight, with equality achieved by the corresponding two-point construction (Dürr, 10 Aug 2025).

The extension from one variable to sums is described as inductive. The proof recursively constructs worst-case distributions for the summands, with each $\mu=\sum_{i=1}^n \mu_i$2 taking either $\mu=\sum_{i=1}^n \mu_i$3 or its maximum $\mu=\sum_{i=1}^n \mu_i$4 subject to the mean constraint, and then uses conditioning and the law of total probability to propagate the extremal structure through the sum. For general $\mu=\sum_{i=1}^n \mu_i$5, the minimal probability is stated to take the recursive form

$\mu=\sum_{i=1}^n \mu_i$6

which leads to the universal lower bound after comparing regimes in $\mu=\sum_{i=1}^n \mu_i$7 and $\mu=\sum_{i=1}^n \mu_i$8 (Dürr, 10 Aug 2025).

The method is explicitly non-asymptotic. It does not appeal to CLT-scale approximations, Berry-Esseen corrections, or higher-moment hypotheses in the final proof, although such tools were central in earlier partial progress (Dürr, 10 Aug 2025).

3. Sharpness, special regimes, and the i.i.d. case

The conjectured lower bound is presented as optimal. In the single-variable case, the exact bound is

$\mu=\sum_{i=1}^n \mu_i$9

and this is attained by the two-point distribution supported on $\delta>0$0 and $\delta>0$1 (Dürr, 10 Aug 2025).

For sums, the paper states that the lower bound remains tight and achievable. One asymptotic regime is especially important: for fixed $\delta>0$2,

$\delta>0$3

This explains the appearance of the constant $\delta>0$4 in the conjecture. By contrast, for small $\delta>0$5, the controlling term is $\delta>0$6 (Dürr, 10 Aug 2025).

The i.i.d. case admits a particularly transparent reduction. A later note gives a short proof for identically distributed random variables by restricting to two-point laws with

$\delta>0$7

equivalently $\delta>0$8, and then rewriting the event $\delta>0$9 as a binomial tail condition. The relevant probability is

$$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$0

and the proof analyzes $$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$1 as a piecewise-decreasing “sawtooth” function. The minimum occurs at $$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$2, giving

$$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$3

for all finite $$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$4 (Egozcue et al., 24 Sep 2025).

This i.i.d. argument is narrower in scope than the full 2025 proof, but it isolates the same extremal phenomenon: sparsity, realized through two-point laws, minimizes the lower-tail mass below the threshold (Egozcue et al., 24 Sep 2025).

4. Earlier bounds and the route to the full proof

Before the general proof, the literature advanced through progressively stronger universal constants. The sequence summarized in the 2025 proof is as follows (Dürr, 10 Aug 2025):

Work	Lower bound / scope	Method
Feige (2006)	roughly $$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$5	reduction to 2-support discrete random variables
Garnett (2018)	$$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$6	higher moments
Guo et al. (2020)	$$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$7	optimization approach and the Berry-Esseen theorem
Alqasem et al. (2024)	conjecture proved for discrete log-concave distributions	restricted distribution class
Egozcue et al. (2025)	conjecture proved for i.i.d. random variables	i.i.d. setting

The 2020 improvement to $$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$8 combined a moment approach based on semidefinite optimization with the Berry-Esseen theorem. In that work, the SDP controls extremal distributions consistent with specified moments, while Berry-Esseen supplies a complementary estimate in large-variance regimes; the final bound is obtained by optimizing over the interaction between these two estimates (Guo et al., 2020).

That hybrid strategy mattered because its two ingredients dominate in different parameter ranges. The moment method is stronger when the variance is small, while Berry-Esseen becomes effective when the variance is large relative to the one-step support constraint. The paper’s best lower bound, $$\mathbb{P}\!\left(\sum_{i=1}^n X_i < \mu+\delta\right)\geq \min\!\left\{\frac{\delta}{1+\delta},\,e^{-1}\right\}.$$9, came from exploiting the fact that the worst cases for the third moment and for the Berry-Esseen term do not coincide (Guo et al., 2020).

The later full proof supersedes these partial constants by establishing the exact conjectured lower bound for all independent non-negative variables with means at most one (Dürr, 10 Aug 2025).

5. Relations to other inequalities and an explicit application

Feige's Conjecture is naturally compared with Markov's inequality. Markov bounds the upper tail of a non-negative random variable: $n$0 whereas Feige's inequality supplies a lower bound on mass below an expectation-plus-$n$1 threshold. The 2025 paper describes this as a converse lower-tail statement under mean constraints, rather than an upper-tail estimate driven by the first moment alone (Dürr, 10 Aug 2025).

The conclusion of the same paper also notes a similarity to Samuels' Conjecture, in that both problems optimize tail probabilities of sums under mean constraints. The resemblance is structural rather than identical: both ask for extremal distributions under sparse assumptions, and both point toward reductions to highly concentrated distribution classes (Dürr, 10 Aug 2025).

An explicit illustration is given in mathematical finance. If one invests in $n$2 stocks and $n$3 denotes the non-negative profit from selling stock $n$4 at time $n$5, with each expected return at most $n$6, then for any $n$7,

$n$8

Equivalently, the probability that the total return exceeds the expected return by $n$9 is at most

$\delta>0$0

The point of the example is that the guarantee does not deteriorate merely because the summands are skewed or heavy-tailed; the inequality remains valid under only independence, non-negativity, and the coordinatewise mean bound (Dürr, 10 Aug 2025).

6. Terminological ambiguity: other conjectures bearing Feige's name

The expression “Feige's Conjecture” is not unique to probability. In extremal combinatorics and CSP theory, it also denotes a 2008 conjecture on even covers in dense $\delta>0$1-uniform hypergraphs. In that setting, an even cover of length $\delta>0$2 is a sequence of $\delta>0$3 hyperedges whose symmetric difference is empty, and the conjecture asserts that every $\delta>0$4-uniform hypergraph on $\delta>0$5 with

$\delta>0$6

hyperedges has an even cover of length $\delta>0$7. A 2021 paper proves this up to a $\delta>0$8 slack in $\delta>0$9 using a spectral double-counting argument based on the Kikuchi matrix and trace-moment methods, and links the result to smoothed Boolean CSP refutation and polynomial-size refutation witnesses below the spectral threshold (Guruswami et al., 2021).

In nonlocal games, “Feige's Conjecture” has also referred to the assertion that Feige's game has no quantum advantage, meaning its quantum value equals its classical value. A 2025 paper disproves that conjecture by showing

$\delta$0

while also proving that for even parallel repetition count $\delta$1,

$\delta$2

The same paper shows that the game is a robust self-test for the $\delta$3-dimensional maximally entangled state (Schmidt et al., 9 Oct 2025).

In hardness of approximation, Feige's name is attached to conjectures about smooth label cover. One 2025 paper describes a classical Feige conjecture asserting NP-hardness for small-gap smooth label cover under regularity and smoothness conditions, and then proves that the quantum smooth label cover problem and the quantum oracularized smooth label cover problem are both RE-hard. Its reductions include a quantum-sound version of Feige's reduction from 3SAT to 3SAT5 (Culf et al., 3 Oct 2025).

Accordingly, in contemporary research usage the phrase requires disambiguation. In probability it denotes the small-deviation inequality proved in 2025; in other areas it refers to distinct conjectures about hypergraph even covers, nonlocal games, and smooth label cover.

7. Status and significance

The probabilistic Feige's Conjecture is now resolved in full generality for arbitrary non-negative independent random variables with means at most one, any $\delta$4, and any $\delta$5 (Dürr, 10 Aug 2025). The proof is constructive in the sense that it identifies extremal distribution classes, reduces the problem to two-point laws, and exhibits the mechanism by which the universal lower bound emerges.

Its significance lies in the strength of the conclusion under weak hypotheses. The theorem provides a universal lower bound below $\delta$6 without invoking variance, subgaussianity, bounded support, log-concavity, or identical distribution. That is why the result is naturally described as a robust anti-concentration principle (Dürr, 10 Aug 2025).

A plausible implication is that the theorem becomes a baseline inequality for settings where only first moments and independence are available. The finance example in the paper illustrates this directly, and the earlier literature cited applications of small-deviation bounds in graph theory and inventory management (Dürr, 10 Aug 2025, Guo et al., 2020).

From a structural viewpoint, the conjecture’s resolution also clarifies the extremal geometry of the problem. The worst cases are not diffuse continuous distributions but sparse two-point laws, and the limiting constant $\delta$7 arises from the large-$\delta$8 behavior of the corresponding recursive extremizers (Dürr, 10 Aug 2025).