##### Department of Mathematics,

University of California San Diego

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### Statistics

## Noureddine El Karoui

#### Stanford University

## The Tracy-Widom law holds when $n, p, p/n ightarrow infty$, with application to PCA

##### Abstract:

Principal Component Analysis (PCA) is a tool used across the spectrum of scientific applications. In modern practice, it is often applied to $n imes p$ data matrices with $n$ and $p$ both large. Classical theory (Anderson 1963) fails to apply in this setting. Using random matrix theory, Johnstone (2000) recently shed light on some theoretical aspects of PCA in this setup. Specifically, when the entries of the $n imes p$ matrix $X$ are iid ${cal N}(0,1)$ and $n/p ightarrow ho in (0,infty)$, he showed that $lambda_{n,p}$ , the largest eigenvalue of the empirical covariance matrix $X'X$, converges to the so-called Tracy-Widom distribution (after proper recentering and rescaling). We will show that the result holds when $n,p ightarrow infty$ and $n/p ightarrow 0$ or $ infty$, in effect removing the need to worry about the limiting behavior of $n/p$. We will also present preliminary results for rates of convergence. Finally, we will illustrate how these and related theoretical insights might be used in practice.

Host: Ian Abramson

### February 26, 2004

### 1:00 PM

### AP&M 6438

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