I would like to calculate the limit value of a linear functional
\begin{equation}
\lim\_{n\rightarrow\infty}\mathcal{I}\_n=\lim\_{n\rightarrow\infty}\frac{1}{n}\sum\_{i=1}^n f(\lambda\_i)=\lim\_{n\rightarrow\infty}\int f(x)\mathrm{d}G\_n(x),
\end{equation}
where $\lambda\_i$ are eigenvalues of a $n\times n$ matrix and $G\_n(x)$ is the empirical eigenvalue distribution function. In the large dimension limit, $G\_n(x)$ converges to a nonrandom limit $G(x)$ but its explicit form is difficult to obtain. In stead, it is easy to obtain the Stieltjes transform of $G(x)$, say $m\_G(x)$.


I hope I could obtain $\lim\_{n\rightarrow\infty}\mathcal{I}\_n$ using $m\_G(x)$. Indeed, the equation (1.14) in
"Z. D. Bai and J. W. Silverstein, CLT for Linear Spectral Statistics of Large-Dimensional Sample Covariance Matrices, Ann. Prob., 2004" shows
\begin{equation}
\int f(x)\mathrm{d}G(x)=-\frac{1}{2\pi i}\int f(z)m\_G(z)\mathrm{d}z.
\end{equation}
But I don't understand how they obtain it and if it applies for general $G(x)$. I'd appreciate someone giving me some hints.



Can I obtain the limit value of a linear spectral statistics using Stieltjes transform?
Let me write the limit value $I$ which you are seeking as


$$I=\int f(x)\rho(x)dx$$


where $\rho=dG/dx$ is the eigenvalue density, with Stieltjes transform $S\_\rho$. The [Stieltjes-Perron inversion formula](http://en.wikipedia.org/wiki/Stieltjes_transformation) reads


$$\rho(x)=\lim\_{\epsilon\rightarrow 0^+}\frac{S\_\rho(x-i\epsilon)+S\_\rho(x+i\epsilon)}{2\pi i}$$


Substitute in the integral over $x$,


$$I=\frac{1}{2\pi i}\lim\_{\epsilon\rightarrow 0^+}\left[\int\_{-\infty-i\epsilon}^{\infty-i\epsilon}f(x)S\_\rho(x)dx-\int\_{-\infty+i\epsilon}^{\infty+i\epsilon}f(x)S\_\rho(x)dx\right]$$


Now assume that $f(x)S\_{\rho}(x)$ is analytic for ${\rm Im}\,x\leq 0$, and decays sufficiently rapidly for ${\rm Im}\,x\rightarrow -\infty$ that the integration contour of the first integral may be closed in the lower half of the complex plane. This integral then evaluates to zero, leaving only the second integral,


$$I=-\frac{1}{2\pi i}\lim\_{\epsilon\rightarrow 0^+}\int\_{-\infty+i\epsilon}^{\infty+i\epsilon}f(x)S\_\rho(x)dx$$


