---
abstract: |
    Let $X$ be a $n\times p$ matrix and $l_1$ the largest eigenvalue of the covariance matrix $X^{*}X$. The “null case" where $X_{i,j}\sim {\cal N}(0,1)$ is of particular interest for principal component analysis.

    For this model, when $n, p{\rightarrow}\infty$ and $n/p {\rightarrow}\gamma \in \mathbb{R}_+^*$, it was shown in @imj that $l_1$, properly centered and scaled, converges to the Tracy-Widom law.

    We show that with the same centering and scaling, the result is true even when $p/n$ or $n/p{\rightarrow}\infty$, therefore extending the previous result to $\gamma \in \overline{\mathbb{R}}_+$. The derivation uses ideas and techniques quite similar to the ones presented in @imj. Following @sosh, we also show that the same is true for the joint distribution of the $k$ largest eigenvalues, where $k$ is a fixed integer.

    Numerical experiments illustrate the fact that the Tracy-Widom approximation is reasonable even when one of the dimension is small.
author:
- |
    Noureddine El Karoui[^1]\
    *Department of Statistics,*\
    *Stanford University*
bibliography:
- 'research.bib'
title: 'On the largest eigenvalue of Wishart matrices with identity covariance when $n$, $p$ and $p/n {\rightarrow}\infty$'
---

Introduction
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Large scale principal component analysis (PCA) - concerning an $n\times p$ matrix $X$ where $n$ and $p$ are both 