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The Sharp R\'enyi and Tsallis Threshold in the Shepp--Olkin Concavity Problem
Haoran Wang
TL;DR
For sums of independent Bernoulli variables, the Shepp–Olkin question concerns joint concavity of uncertainty, with q = 1 the sharp boundary between concavity below and failure above. For orders below one, the proof combines the Hillion–Johnson transport inequality with an explicit nonlinear telescoping correction and a Riccati identity with a zero-crossing argument. Both Rényi and Tsallis entropies are jointly concave exactly for 0 < q ≤ 1; for q > 1, a transverse two-coin path gives strict local convexity.
Problem
For sums of independent Bernoulli variables, the Shepp–Olkin question concerns joint concavity of uncertainty, with q = 1 the sharp boundary between concavity below and failure above.
Source: §1 · pp. 1–2Method
For orders below one, the proof combines the Hillion–Johnson transport inequality with an explicit nonlinear telescoping correction and a Riccati identity with a zero-crossing argument.
Source: §1.3 · p. 3, §4 · p. 9Results
Both Rényi and Tsallis entropies are jointly concave exactly for 0 < q ≤ 1; for q > 1, a transverse two-coin path gives strict local convexity.
Source: §1.2 · pp. 2–3Takeaways & Limitations
The above-one obstruction is transverse, showing that monotone or one-parameter tests do not locate the joint-concavity boundary.
Source: §4 · p. 9The stated setting is sums of independent Bernoulli variables, and the argument first treats interior parameter points.
Source: §1.2 · p. 2, §2 · p. 4
Abstract
from arXiv · showhide
Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\ldots,p_n$, and let $S=\sum_i B_i$. Hillion and Johnson proved that the Shannon entropy of $S$ is jointly concave in the parameter vector and proposed corresponding critical-order conjectures for R'enyi and Tsallis entropies, with predicted thresholds $2$ and approximately $3.65986$, respectively. We determine both thresholds exactly. For every $0<q<1$, the power sum $\sum_k \mathbb P(S=k)^q$ is jointly concave in $(p_1,\ldots,p_n)$, and strictly concave on the open parameter cube. Consequently, the R'enyi and Tsallis entropies of order $q$ are jointly concave. At $q=1$ this agrees with the Shannon theorem. For every $q>1$, joint concavity fails already for the sum of two Bernoulli variables: a transverse interpolation in which the two parameters move in opposite directions gives strict local convexity for both entropies. Hence the universal joint-concavity range for both families is exactly $0<q\leq 1$. Below order one, the proof combines the Hillion--Johnson transport inequality with an explicit nonlinear telescoping correction. The corrected local curvature reduces to a two-dimensional quadratic form. An exact Riccati identity, together with a one-sided zero-crossing argument, proves positivity of its determinant throughout the full range $0<q<1$.
1 Introduction
The paper revisits joint entropy concavity for sums of independent Bernoulli variables and determines the exact Rényi and Tsallis order range. It develops a nonlinear correction to the prior transport approach to establish the sub-Shannon result.
1.1 Prior work
- Hillion and Johnson proved the Shepp–Olkin conjecture that Shannon entropy is jointly concave in (p1, …, pn), and conjectured Rényi threshold 2 and Tsallis threshold q ≈ 3.65986.Source: §1 · pp. 1–2
1.3 Technique overview
- The earlier telescoping-derivative suggestion is developed into an explicit nonlinear correction, paired with a Riccati identity and a zero-crossing argument.The correction reduces local curvature to a two-dimensional quadratic form whose determinant is shown positive throughout 0 < q < 1.Source: §1.1 · p. 2, §1.3 · p. 3
1.1 Prior work
- The above-one obstruction is transverse, unlike monotonicity tests that move parameters toward fairer coins, so monotone tests do not locate the joint-concavity threshold..Source: §1.1 · p. 2
1.2 Our results
- Both Rényi and Tsallis entropies are jointly concave exactly for 0 < q ≤ 1, with strict concavity on the open cube when 0 < q < 1.At q = 1 they reduce to Shannon entropy; above one, a two-coin interior path gives both entropies a positive second derivative at its midpoint.Source: §1.2 · pp. 2–3
- For q > 1, the two-variable obstruction embeds into every higher-dimensional parameter cube, so joint concavity on [0, 1]^n fails for every n ≥ 2.Source: §1.2 · p. 3
2 Preliminaries
The proof works along interior affine paths in the Bernoulli parameter cube, then extends conclusions to boundary points by continuity. It encodes changes in the sum’s probability law through probabilities with one or two Bernoulli variables removed.
2 Preliminaries
- Boundary points follow by continuity after the argument is carried out at interior parameter points along affine paths.The path is written as pi(t) = pi + tbi.Source: §2 · p. 4
- Probabilities with one or two Bernoulli variables removed define the auxiliary mass functions used in differentiating the sum’s law.The one-variable removal probability is f^(i)_k(t); the two-variable removal probability is f^(i,j)_k(t).Source: §2 · p. 4
- Strict log-concavity of the interior Poisson-binomial law lets the transport argument control the second-order coefficient hk using neighboring first-order coefficients gk and gk+1.Along a general affine direction, hk has no fixed sign.Source: §2 · p. 4
3 Proof of the sharp range
Above order one, both entropy families fail joint concavity; below one, a nonlinear telescoping correction reduces the curvature proof to positivity of local quadratic forms. The argument establishes strict concavity on the open cube and extends joint concavity to the closed cube.
3.1 All orders above one fail
- For every q > 1, both entropy families fail joint concavity.The proof uses a two-coin path; the section’s displayed probability expressions yield strict local convexity.Source: §3.1 · pp. 4–5
3.2 A nonlinear telescoping correction below one
- For 0 < q < 1, the telescoping correction and transport bound reduce each local curvature term to a two-by-two quadratic form whose determinant must be positive.The correction is antisymmetric and homogeneous of degree −1 − s, with a continuous extension to the diagonal.Source: §3.2 · pp. 5–7
- A boundary expansion and one-sided zero-crossing argument establish determinant positivity throughout ac < 1.The determinant is positive just inside the boundary; any interior zero would have the wrong crossing direction.Source: §3.2 · pp. 7–8
- For 0 < q < 1, the power sum is jointly concave on the closed parameter cube and strictly concave on its interior.The proof first establishes curvature on the interior, then uses continuity at the boundary.Source: §3.2 · p. 8
- For 0 < q < 1, Tsallis entropy is jointly concave and strictly concave on the open cube.Source: §3.2 · p. 8
4 Discussion
The obstruction above order one is transverse, already visible with two coins, while the proof below one modifies the Shannon-era transport approach. The paper identifies extensions to other convolution families and quantitative stability as open directions.
4 Discussion
- The above-one obstruction varies two Bernoulli parameters in opposite directions, so one-parameter or monotone tests do not locate the threshold.The discussion contrasts these mixed-sign directions with monotonicity tests toward fairer coins.Source: §4 · p. 9
- Below one, a nonlinear correction redistributes neighboring quadratic terms so the transport bound yields a positive two-by-two form.A boundary expansion and one-sided zero-crossing rule establish the determinant’s positivity across 0 < s < 1.Source: §4 · p. 9
- The paper leaves open whether the correction–Riccati mechanism extends to other ultra-log-concave convolution families and whether explicit Hessian lower bounds can be obtained.The proposed stability bounds would depend on the parameter vector and interpolation direction.Source: §4 · p. 9
A Algebraic details for the local determinant
The algebraic details expose cancellations in the boundary expansion and derive the zero-crossing calculation used for the local determinant argument.
A Algebraic details for the local determinant
- The reciprocal identity cancels the singular 1/δ contribution, while the Riccati identity cancels the constant term, leaving a linear first nonzero coefficient.The calculation concerns the inward curve approaching ac = 1.Source: A Algebraic details for the local… · p. 10
- At a zero of D, eliminating A_s(c) yields the stated derivative identity, and z − s > 1 + s(1 − z) supplies its strict sign.The passage states that no inequality is used before this final sign step.Source: A Algebraic details for the local… · p. 10
B Boundary parameters
For fixed q in (0,1), continuity on the closed parameter cube lets joint concavity extend from interior parameters to boundary points. The same extension applies to Rényi and Tsallis entropy.
B Boundary parameters
- For fixed q in (0,1), the map under discussion is continuous on the closed parameter cube.Source: B Boundary parameters · p. 10
- Approximating boundary points by interior points and passing to the limit extends joint concavity to the closed cube.Source: B Boundary parameters · p. 10
- The same argument extends concavity to Rényi and Tsallis entropy because the power sum is positive and both outer functions are continuous on its range.Source: B Boundary parameters · p. 10