Matrix p-Chebyshev inequality (conjecture)
Establish a matrix p-Chebyshev inequality for i.i.d. random symmetric matrices X_1, …, X_n ∈ S_d with mean M and central p-th moment matrix V_p := E[(abs(X_1 − M))^p] for p ∈ [1,2], by proving the existence of a function f: [1,2] × N → (0,∞) that is bounded in p and sublinear in d such that E[(abs(X_1 + ··· + X_n − nM))^p] ⪯ n f(p,d) V_p, and consequently P(abs( X̄_n − M ) ⪯ A) ≤ n^{1−p} f(p,d) tr(V_p A^{−p}) for all A ∈ S_d^{++}.
References
The following conjecture is an ideal extension of \cref{cor:vec-p,cor:scl-p}. Let $X_1, \dots, X_n$ be i.i.d.\ random matrices taking values in $S_d$ with common mean matrix $\Exp X_1 = M$ and $p$\textsuperscript{th} central moment matrix $ V_p := \Exp (abs(X_1 - M))p$. Then, for any $A \in S_d{++}$, there exists a function $f: [1,2] \times \mathbb N \to (0,\infty)$ that grows sublinearly in its second argument, and bounded in its first argument, such that the $p$\textsuperscript{th} central moment contracts, \begin{equation} \Exp (abs(X_1 + \dots + X_n - n M)p ) n \cdot f(p,d) \cdot V_p; \end{equation} and consequently, for $\overline{X}_n = \frac{1}{n}(X_1 + \dots + X_n)$, \begin{equation}\label{eqn:pumci} \Pr( abs(\overline{X}_n - M) A ) n{1-p} \cdot f(p, d) \cdot tr( V_p A{-p} ). \end{equation}
eqn:pumci: