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I am working on some open problem. And I have reduced the original problem to the "conjecture" (actually I am not familiar with random matrix theory or other fields that may have such a result) as follows.

Consider a random variable $X$ supported on $\mathbb{R}^{d}$ with distribution $P_X$. Assume we have constructed a cover of $\mathbb{R}^{d}$ denoted as $\{\Lambda(\omega):\omega\in\Omega\}$, where $\Omega$ is a finite set. We then make $\Omega$ into some probability space and introduce an extra r.v. $\sigma$ that takes value in $\Omega$.

Now since we have (or not, why?)

$$ \mathbb{E}_{\sigma}\int_{\Lambda(\sigma)}XX^{T}dP_{X} = \mathbb{E}_{X}[XX^{T}] $$,

there should exist some constant $\lambda >0$, such that the following operator inequality is true with probability (in the order of positive semi-definiteness)

$$ P_{X}(\Lambda(\sigma))\mathbb{E}_{X}[XX^{T}] - \lambda{I} \preceq \int_{\Lambda(\sigma)}XX^{T}dP_{X}\preceq P_{X}(\Lambda(\sigma))\mathbb{E}_{X}[XX^{T}] + \lambda{I} $$

From my perspective, the existence may be easy as we could somehow add some additional assumptions on $P_X$ to assert the "boundedness" of $XX^{T}$. Thus we could take $\lambda$ larger than the upper bound.

Could anyone provide some reference or hints relevant to this problem? That will help a lot.

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  • $\begingroup$ 1°) Your first identity needs the "cover" to be a "partition" of $\mathbb R^d$. 2°) As it is, the question has the trivial answer $\lambda=0$ (a sum of two positive semi-definite matrices, such as $XX^T$ and $YY^T$, is greater than each of them). 3°) You probably want something else, as maybe the conditional expectation of $XX^T$ given that $X\in\Lambda(\omega)$ is $\preceq\mathbb E[XX^T]+\lambda I$, don't you? That will depend on the partition and the distribution of $X$. $\endgroup$ – Jean Duchon Mar 31 '18 at 14:08
  • $\begingroup$ Thank you for comments. That's true I am more concerned about the second inequality. I am trying to find out some uniform $\lambda$ that can bound such an operator inequality for any value of $\sigma$. And you are right even the estimation of the uniformly correct $\lambda$ should be based on the structure of the given partition (and about the distribution of X, I'm hoping quite slight constraints on it). The structure of $\Lambda(\sigma)$ is over-complicated and problem-specific and that is why I have not put it here. So is there any reference you could come up with as hints on tackling it? $\endgroup$ – Morino_Hikari Apr 1 '18 at 2:04

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