Expectation of inverse of random matrices

Question

Assume that $\mathbf{X}$ is a random positive-definite matrix. Then, is there any upper or lower bound on the expectation of the following expression $$\mathbb{E}[\mathbf{X}^{-1}]-\alpha\mathbb{E}[\mathbf{X}^{-2}]$$ based on $\mathbb{E}[\mathbf{X}]$?

Answer 1

Let us assume that $\alpha>0$. Then, by rescaling, without loss of generality $\alpha=1$. So, we have to provide an upper or lower bound on $Ef(X)$, where $X$ is a random $n\times n$ positive-definite matrix with a given mean $EX$ and $$f(x):=\frac1x-\frac1{x^2}$$ for real $x>0$.

First of all, there is no finite lower bound here. Indeed, already for $n=1$, letting $P(X=t)=1/2=P(X=2-t)$ with $t\downarrow0$, we get $Ef(X)\to-\infty$.

However, we can get an upper bound on $Ef(X)$, which will be exact if for some $b\in(0,2]$ we have $P(X=bI)=1$, where $I$ is the $n\times n$ identity matrix. Indeed, $$f'(x)=\frac{2-x}{x^3},\quad f''(x)=2\frac{x-3}{x^4},$$ so that $f$ is increasing on $(0,2]$, decreasing on $[2,\infty)$, and concave on $(0,2]$. So, for any $a\in(0,2]$ and all real $x>0$, $$f(x)\le g_a(x):=f(a)+f'(a)(x-a).$$ So, by the spectral decomposition, $f(X)\le g_a(X)$, and hence $$Ef(X)\le Eg_a(X)=B_a(EX):=(f(a)-f'(a)a)I+f'(a)EX.$$ The upper bound $B_a(EX)$ on $Ef(X)$ will be exact if $P(X=bI)=1$ for some $b\in(0,2]$ and $a=b$.

More generally, if e.g. $EX\le mI$ for some real $m>0$, then $$Ef(X)\le\min_{a\in(0,2]}B_a(mI) =\begin{cases} \tfrac14\,I &\text{ if }m\ge2,\\ \tfrac{m-1}{m^2}\,I &\text{ if }0<m\le2. \end{cases}$$

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If now $\alpha\le0$, then the corresponding function $$F(x):=\frac1x-\frac\alpha{x^2}$$ is convex, and hence $EF(X)\ge F(E(X)$ by Jensen's inequality. In this case, there is no upper bound, though -- as the above example with $n=1$, $P(X=t)=1/2=P(X=2-t)$, and $t\downarrow0$ shows.