# Expected operator norm of inverse Wishart matrix

Let $W\sim W_p(n,I)$ be a white $p\times p$ Wishart matrix, and assume $n>p+1$, which ensures that $W$ is invertible almost surely. Let $\|W^{-1}\|_{\text{op}}$ be the operator norm (maximum eigenvalue) of $W^{-1}$. Assume also that $p/n \to c<1$ for some constant $c$.

I'd like to get an upper bound on the expected norm $E\:\|W^{-1}\|_{\text{op}}$, and hopefully show that this number is $O(n^{-1})$, or $o(n^{-1/2})$ as $n,p\to \infty$.

Note that the expectation of $\|W^{-1}\|_{\text{op}}$ is finite for any fixed $n$ and $p$, since the operator norm is upper bounded by $\text{trace}(W^{-1})$, and the expected value of the trace is known. (see e.g. the book of Kollo and von Rosen, 2005)

Note also that $\|W^{-1}\|_{\text{op}}$ is the reciprocal of the smallest eigenvalue of $W$.

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Unless I am mistaken (which is quite possible), in the other direction, it seems relatively easy to establish that $\mathbf E \|W^{-1}\|_{\mathrm{op}} \geq n^{-1}$ and this seems to hold regardless of the relationship between $n$ and $p$, so one can't hope for much better than what you've stated. – cardinal Jan 22 '12 at 20:56
If you aren't familiar with it, you should also look at R. Vershynin's recent Introduction to the non-asymptotic analysis of random matrices, particularly subsection 5.3. – cardinal Jan 22 '12 at 21:01