# Is $\mathcal{P}= \{ P: E[XX^T] \preceq A\}$ is sequentially compact?

by Lisa   Last Updated January 12, 2018 20:20 PM

Let $\mathcal{P}$ be a set of probability distributions on $\mathbb{R}^n$ such that \begin{align} \mathcal{P}= \{ P: E[XX^T] \preceq A\}. \end{align} where $A$ is a positive-definite matrix and where the expectation is take with respect to $P$. Also, $B \preceq A$ implies that $A-B$ is positive semi-definte matrix.

Question: Is the set $\mathcal{P}$ sequentially compact?

For $n=1$, $\mathcal{P}= \{ P: E[X^2] \le A\}$ is tight by Markov inequality and is therefore tight by Prohorov's theorem.

However, I am not sure what tools to use for $n>1$. Is there a way to prove tightness in this case?

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