Hello,


I'm interested in the distribution of the trace of an inverse-Wishart matrix $W\_n^{-1}(I,n)$, where $I$ is $n\times n$ identity matrix. More precisely, I seek for an asymptotic estimate (when $n\to\infty$) for a function $f(n)$ such that $Pr[Tr(W)< f(n)]>2/3$, say.


What I've learned so far:


1. I know the pdf of the eigenvalues of $W$, thus, I could use it to find the bound, but it could be tedious.
2. A formula for the expectation of $W\_n^{-1}(I,m)$ is known, but it does not work for $n=m$ (the expectation is infinite in this case).


This question seems pretty basic, so I expect it should have been considered before, but my google search hasn't revealed any reference on it so far. Could you help me with any?


Thank you,


Alexander.



Distribution of trace of inverse-Wishart matrix $W_n(I,n)$
I guess, I've done it.


My idea is to detach two eigenvalues, use the smallest eigenvalue estimation for them and then apply the known result for the expectation of the rest of the matrix.


Let $G$ be $n\times n$ matrix with entries being pairwise independent standard Gaussians. Then let $W=G^TG$ and I am interested in $Tr(W^{-1})$. I use that mine multivariate Gaussian distributions are invariant under application of (real) orthogonal matrices. Thus, the distribution does not change if I assume that $e\_1$ and $e\_2$ (elements of the standard basis) are eigenvectors of $W$. This means, merely, that first two columns of $G$ are orthogonal to all other columns and orthogonal to each another. Again, as orthogonal transformations does not change the distribution on $G$, this assumption does not change the distribution on the rest of the columns of $G$. Let $\lambda\_1$ and $\lambda\_2$ be the eigenvalues of $e\_1$ and $e\_2$, respectively, and let $W'$ be $W$ with first two rows and columns removed. Then $Tr(W^{-1}) = 1/\lambda\_1 + 1/\lambda\_2 + Tr(W'^{-1})$. I can use the estimation on the smallest eigenvalue of $W$ to bound the first two terms. Now note that $W'$ is, in fact, $W\_{n-2}(I,n)$, and it is known that expectation of $W'^{-1}$ is $I$. Hence, the expectation of the trace is $n-2$. Using Markov for the third term, I get that, with high enough probability, $Tr(W^{-1})$ is $O(n)$. This cannot be improved because of the smallest eigenvalue.


I hope that's correct.


