Let mathbfX_1,…,mathbfX_n be i.i.d. matrices with common mean matrix EmathbfX_1=mathbfM and pth central moment matrix mathbfV_p:=Ebigl(operatornameabs(mathbfX_1-mathbfM)bigr)^p. For any mathbfA∈S_d^++, Matrix p-Chebyshev inequality. There exists a function f:[1,2]×N→(0,∞) that grows sublinearly in its second argument and is bounded in its first argument, such that Ebigl(operatornameabs(mathbfX_1+…+mathbfX_n-n mathbfM)^p bigr)preceq n,f(p,d),mathbfV_p; and, consequently, for overlinemathbfX_n=frac1n(mathbfX_1+…+mathbfX_n), Pbigl(operatornameabs(overlinemathbfX_n-mathbfM)npreceq mathbfAbigr)leqslant n^1-pf(p,d)operatornametr(mathbfV_p mathbfA^-p). This is proposed as an ideal matrix extension of the scalar and vector p-Chebyshev inequalities. The source provides no resolution, so the conjecture remains open.

Matrix p-Chebyshev inequality

From papers

Let $\mathbf{X}_1,\dots,\mathbf{X}_n$ be i.i.d. matrices with common mean matrix $\mathbb{E}\mathbf{X}_1=\mathbf{M}$ and $p$ th central moment matrix

\mathbf{V}_p:=\mathbb{E}\bigl(\operatorname{abs}(\mathbf{X}_1-\mathbf{M})\bigr)^p.

For any $\mathbf{A}\in\mathcal{S}_d^{++}$ , Matrix $p$ -Chebyshev inequality. There exists a function $f:[1,2]\times\mathbb{N}\to(0,\infty)$ that grows sublinearly in its second argument and is bounded in its first argument, such that

\mathbb{E}\bigl(\operatorname{abs}(\mathbf{X}_1+\dots+\mathbf{X}_n-n\mathbf{M})^p\bigr)\preceq n\,f(p,d)\,\mathbf{V}_p;

and, consequently, for $\overline{\mathbf{X}}_n=\frac{1}{n}(\mathbf{X}_1+\dots+\mathbf{X}_n)$ ,

\mathbb{P}\bigl(\operatorname{abs}(\overline{\mathbf{X}}_n-\mathbf{M})\npreceq\mathbf{A}\bigr)\leqslant n^{1-p}f(p,d)\operatorname{tr}(\mathbf{V}_p\mathbf{A}^{-p}).

This is proposed as an ideal matrix extension of the scalar and vector $p$ -Chebyshev inequalities. The source provides no resolution, so the conjecture remains open.

Progress summary

Open

No verified proof or disproof of the proposed matrix extension has been found, so the problem remains open.

Wang and Ramdas proposed this matrix analogue in 2024: independent centered matrix sums should satisfy a dimension-sublinear moment bound, yielding concentration for their empirical mean. The retrieved literature does not resolve this exact assertion.

Known results

Wang and Ramdas (2024): an exchangeable positive-semidefinite matrix $p$ -Chebyshev tail bound using raw moments.
Wang and Ramdas (2024): a trace/operator-norm bound for centered exchangeable matrices using $\operatorname{tr}(\mathbf{V}_p)$ .
A separate 2024 preprint gives randomized matrix Chebyshev and Chernoff inequalities for $1\leq p<2$ , but not the stated i.i.d.-sum Loewner-order bound.

Current status (as of August 2026): The proposed i.i.d.-sum inequality and its concentration consequence remain unproved and undisproved; only related single-matrix or exchangeable-sequence inequalities are documented.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Hongjian Wang and Aaditya Ramdas, “Positive Semidefinite Matrix Supermartingales”, arXiv:2401.15567 (2025).

1 week ago

Source-version clarification: this statement appears as Conjecture A.4 only in versions 1–3 of Wang and Ramdas, https://arxiv.org/html/2401.15567v3#A1.SS2 . The conjecture was removed beginning with version 4 in January 2025 and is absent from the current version 6. The argument below addresses the withdrawn historical claim still reproduced on this problem page; it does not challenge the distinct valid results in the current paper.

In fact, for every fixed 1≤p<2, there is no finite distribution-independent constant f(p,2), even in dimension d=2 with n=2 bounded positive-semidefinite observations.

Fix 0<θ<π/2, let s=sin θ, and define rank-one orthogonal projections

Q=e₁e₁ᵀ, R=vvᵀ, v=(cos θ,sin θ).

Let Y be uniformly distributed on {Q,−Q,R,−R}, and let X=I+Y. Then 0≼X≼2I, its mean is M=I, and projection idempotence gives

V_p=E|X−M|^p=E|Y|^p=(Q+R)/2.

For independent copies Y₁,Y₂, the four ordered pairs

(Q,−R), (−R,Q), (−Q,R), (R,−Q)

have combined probability 1/4. Since

(Q−R)²=s²I,

each contributes |Y₁+Y₂|^p=s^pI. All other terms are positive semidefinite; therefore

E|Y₁+Y₂|^p≽(s^p/4)I.

Testing the conjectured matrix inequality

E|Y₁+Y₂|^p≼2f(p,2)V_p

against e₂ yields

s^p/4≤f(p,2)s²,

and consequently

f(p,2)≥s^{p−2}/4.

As θ↓0, this lower bound diverges for every 1≤p<2. Hence no finite dimension-two constant exists; allowing arbitrary dimension dependence cannot repair the claim.

There is also a fully rational certificate at p=1. For each integer m≥1 set

v=((m²−1)/(m²+1),2m/(m²+1)), R=vvᵀ,

and retain the same four-point law. Exact enumeration gives

E|Y₁+Y₂|=V₁+[m/(2(m²+1))]I.

For w=(−1,m), one has

wᵀV₁w=1, wᵀE|Y₁+Y₂|w=1+m/2,

so the claimed bound forces

f(1,2)≥1/2+m/4

for every m. At p=2 the usual variance identity does hold with f(2,2)=1; the obstruction concerns precisely the strictly subquadratic range.

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