Let $\mathbf{X}\in\mathbb{R}^n$ be a vector random variable, and let $\mathbf{P}$ be an $n\times n$ positive semi-definite matrix.
I'm interested in deriving lower bounds on the expected value of the quadratic form $\mathbf{X}^\top\mathbf{P}\mathbf{X}$ that depend only on the mean of $\mathbf{X}$ rather than higher moments. One such bound can be derived by noting that $\mathbf{X}^\top\mathbf{P}\mathbf{X}$ is a convex function of $\mathbf{X}$, and applying Jensen's inequality to get $$E[\mathbf{X}^\top\mathbf{P}\mathbf{X}]\geq E[\mathbf{X}]^\top\mathbf{P}E[\mathbf{X}].$$ I'm wondering what lower bounds can be derived that don't depend on the off-diagonal elements of $\mathbf{P}$.
Two such examples for specific cases are as follows:
If $\mathbf{P}$ is diagonal, then we have $$E[\mathbf{X}^\top\mathbf{P}\mathbf{X}]\geq \sum_{i,j}P_{ij}E[X_i]^2.$$
If all elements of $\mathbf{P}$ are non-negative, and $E[X_i] = E[X_1]\geq0$ for all $1\leq i\leq n$, then $$E[\mathbf{X}^\top\mathbf{P}\mathbf{X}]\geq \mathrm{Tr}(\mathbf{P})E[X_1]^2.$$
Are there any more general lower bounds that can be derived for $E[\mathbf{X}^\top\mathbf{P}\mathbf{X}]$ that depend only on $E[\mathbf{X}]$ and the diagonal elements of $\mathbf{P}$?
Thanks!
If $E[X_iX_j]=0$ for all $i\ne j$ (e.g. when the $X_i$s are uncorrelated RVs with zero means), then $$ E[X^\top PX]=E[\operatorname{tr}(PXX^\top)]=\operatorname{tr}(PE[XX^\top])=\sum_iP_{ii}E[X_i^2]. $$