0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 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,

$$I=-\frac{1}{2\pi i}\lim_{\epsilon\rightarrow 0^+}\int_{-\infty+i\epsilon}^{\infty+i\epsilon}f(x)S_\rho(x)dx$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.