Iterated moment minor inequalities

Simić, Moment inequalities of the second and third order, §2:

Conjecture 1 The inequality φ(r₁,s₁;u₁,v₁)φ(r₂,s₂;u₂,v₂) ≥ [(ξ((r₁+r₂)/2,(s₁+s₂)/2) − ξ((r₁+s₂)/2,(s₁+r₂)/2))(ξ((u₁+u₂)/2,(v₁+v₂)/2) − ξ((u₁+v₂)/2,(v₁+u₂)/2)) − (ξ((r₁+u₂)/2,(s₁+v₂)/2) − ξ((r₁+v₂)/2,(s₁+u₂)/2))(ξ((u₁+r₂)/2,(v₁+s₂)/2) − ξ((u₁+s₂)/2,(v₁+r₂)/2))]² holds for arbitrary r_i, s_i, u_i, v_i ∈ I, i ∈ {1, 2}?

With σ_a(x) := ξ(x, x+a) and θ_a(x,y) := σ_a(x)σ_a(y) − σ_a²((x+y)/2):

Conjecture 2 Is it true that θ_a(x,y)θ_a(z,v) ≥ [θ_a((x+z)/2,(y+v)/2) − θ_a((x+v)/2,(y+z)/2)]²?

For a positive random variable X with mean m, the definitions are

$$\lambda_s=\begin{cases} \log m-\mathbb E\log X,&s=0,\ \mathbb E(X\log X)-m\log m,&s=1,\ \dfrac{\mathbb E X^s-m^s}{s(s-1)},&s\ne0,1, \end{cases}\qquad \xi(s,t)=\lambda_s\lambda_t-\lambda_{(s+t)/2}^2,$$

$$\varphi(s,t;u,v)=\xi(s,t)\xi(u,v) -\left[\xi!\left(\frac{s+u}{2},\frac{t+v}{2}\right) -\xi!\left(\frac{s+v}{2},\frac{t+u}{2}\right)\right]^2.$$

Formal reading

All Lean names below have prefix D5.S3.Analytic.MomentMinors.

| Source object | Lean name | Correspondence | | --- | --- | --- | | Positive probability law of X | Defs.Admissible μ I | A Borel probability measure on ℝ, positive almost everywhere, with integrable identity. | | Moments exist for every s ∈ I | Defs.Admissible.integrable_rpow | Integrability of each real power; log X at 0 ∈ I and X log X at 1 ∈ I are separate conditions. | | Interval I | I.OrdConnected | Any order-connected real set, with all needed midpoints still in I. | | λ_s, including s = 0,1 | Defs.lam, Defs.mean | Exactly the displayed three-case definition, with m the integral of the identity. | | ξ, φ | Defs.xi, Defs.phi | The displayed second and iterated second minors. | | σ_a, θ_a | Defs.sigma, Defs.theta | The displaced pair and its midpoint second minor. | | Conjecture 1 | Defs.claim1 | All eight endpoint parameters independently range over I, in the quoted index order. | | Conjecture 2 | Defs.claim2 | Every real a,x,y,z,v such that x,y,z,v and x+a,y+a,z+a,v+a belong to I. No sign restriction on a. | | Both assertions | Defs.claim, Result.result | The conjunction, for every interval and every admissible law. |

The settling declaration is D5/S3/Analytic/MomentMinors/Result.result. The source note is D5/L/Divergence/simic2018momentinequalities. Both inequalities refine the nonnegativity of moment minors by controlling cross terms from independently chosen parameter pairs.

Status: Proved for every admissible law and all parameters in the stated interval. The logarithmic cases, repeated parameters, vanishing minors and negative displacements are included.

Set K(s,t) = λ_{(s+t)/2}. For s different from zero and one, use

$$f_s(x)=\frac{x^s-sx+s-1}{s(s-1)}.$$

At zero and one use f_0(x) = x−1−log x and f_1(x) = x log x−x+1. In every case f_s''(x) = x^{s−2} on x > 0 and λ_s = E f_s(X)−f_s(E X). For any finite family t_i ∈ I and real c_i,

$$F(x)=\sum_{i,j}c_ic_jf_{(t_i+t_j)/2}(x),\qquad F''(x)=\left(\sum_i c_i x^{t_i/2-1}\right)^2\ge0.$$

The mean m is positive. Integrating the supporting tangent inequality F(m)+F'(m)(x−m) ≤ F(x) gives Σ_{i,j} c_i c_j λ_{(t_i+t_j)/2} ≥ 0. All potentials are integrable by the given moment assumptions; no derivative integrability is needed. Thus every finite matrix of K is positive semidefinite.

For a finite positive semidefinite real matrix A and arbitrary ordered index pairs (p(a),q(a)), put

$$C_{ab}=A_{p(a)p(b)}A_{q(a)q(b)}-A_{p(a)q(b)}A_{q(a)p(b)}.$$

Let T have column e_{(p(a),q(a))}−e_{(q(a),p(a))}. Then C = Tᵀ(A⊗A)T/2, so C is positive semidefinite. For K this gives

$$B((r,s),(u,v))= \xi!\left(\frac{r+u}{2},\frac{s+v}{2}\right) -\xi!\left(\frac{r+v}{2},\frac{s+u}{2}\right).$$

The two central squared λ terms cancel because their arguments agree. The diagonal is ξ(r,s). Applying the same construction to B on the four pairs (r₁,s₁),(u₁,v₁),(r₂,s₂),(u₂,v₂), and taking its two by two determinant, gives Conjecture 1. Restricting B to pairs (x,x+a) gives the midpoint kernel σ_a((x+y)/2). Its second minors and their two by two determinant give Conjecture 2; the two central squared σ terms again cancel.

The same construction continues to every finite depth in prose: start with K_0 = K and define K_{n+1}((r,s),(u,v)) = K_n(r,u)K_n(s,v)−K_n(r,v)K_n(s,u). Induction gives positive semidefiniteness of every finite restriction: at each stage collect the finitely many endpoints and apply the arbitrary-pair lemma. Its principal two by two determinants give further inequalities. This induction is not asserted as an additional Lean theorem.

For finite strictly positive probability vectors p and q, let X take p_i/q_i with probability q_i. Then E X = 1, every real power and both logarithmic moments are integrable, and

$$\lambda_s=\frac{\sum_i p_i^s q_i^{1-s}-1}{s(s-1)}\quad(s\ne0,1),\qquad \lambda_0=D(q\Vert p),\quad\lambda_1=D(p\Vert q).$$

These are the f-divergences for generators f_s above, since the affine terms have zero expectation. Substitution in the two inequalities gives inequalities among these divergences, including the two relative entropies. This discrete correspondence is a consequence in prose, not an additional Lean theorem.

An admissible law and eight parameters in its interval violating Conjecture 1, or an admissible law and a,x,y,z,v satisfying all eight endpoint conditions but violating Conjecture 2, would contradict the statement. Laws lacking the specified integrability or positivity conditions are outside its scope.

D5/S3/Analytic/MomentMinors/Result.result proves exactly Defs.claim, without additional premises. Its principal dependencies are MomentKernel.momentKernel_posSemidef and PairMinor.pairMinor_posSemidef. The former integrates a supporting tangent for a finite quadratic combination of the normalized power potentials. The latter applies Matrix.PosSemidef.kronecker, Matrix.PosSemidef.conjTranspose_mul_mul_same and nonnegative scaling. Successive specializations and Matrix.PosSemidef.det_nonneg with Matrix.det_fin_two give the two conjuncts. The axiom closure of Result.result is {propext, Classical.choice, Quot.sound}.

The proof route was derived with the assistance of the ChatGPT Pro model. The source proves φ ≥ 0, log-convexity of σ_a and restricted three- and four-parameter cases of Conjecture 1. The journal version states the eight-parameter refinement as an open problem. No settlement was found in the supplied literature scope: the arXiv versions, the titles, identifier, DOI and author in conjunction with conjecture, proof and counterexample, and the OpenAlex citation record. The identified citing work by Sason concerns lower-order f-divergence refinements. This scope does not establish exhaustive citation-index coverage or priority.

• PROVED. Conjectures 1 and 2 for every admissible law and all parameters in I, by Result.result.
• PROVED. The moment kernel (s,t) ↦ λ_{(s+t)/2} is positive semidefinite on every finite family in I, by MomentKernel.momentKernel_posSemidef.
• PROVED. Matrices of two by two minors of a positive semidefinite real matrix along arbitrary index pairs are positive semidefinite, by PairMinor.pairMinor_posSemidef.
• PROVED in prose. Iteration to every finite depth preserves positive semidefiniteness, by induction using the arbitrary-pair construction.
• PROVED in prose. The finite discrete specialization X = p_i/q_i under q gives inequalities among f-divergences, with λ_0 and λ_1 equal to the reverse and forward relative entropies, respectively, as derived above.

Exhaustive literature coverage, citation-index completeness and priority are unverified. Independent external-source confirmation of the supplied bounded literature findings is unverified.