Let $\mathbb{ A}_p=\frac{ \mathbb{ Y}\mathbb{ Y}^*}{ m}$ and $\mathbb{ B}_p=\frac{ \mathbb{ X}\mathbb{ X}^*}{ n}$ be two independent random matrices where $\mathbb{ X}=(X_{ ij})_{ p \times n}$ and $\mathbb{ Y}=(Y_{ ij})_{ p \times m}$ respectively consist of real (or complex) independent random variables with $\mathbb{ E}X_{ ij}=\mathbb{ E}Y_{ ij}=0$, $\mathbb{ E}|X_{ ij}|^2=\mathbb{ E}|Y_{ ij}|^2=1$. Denote by $\lambda_{ 1}$ the largest root of the determinantal equation $\det(\lambda \mathbb{ A}_p-\mathbb{ B}_p)=0$. We establish the Tracy-Widom type universality for $\lambda_{ 1}$ under some moment conditions on $X_{ ij}$ and $Y_{ ij}$ when $p/m$ and $p/n$ approach positive constants as $p\rightarrow\infty$.
Categories: math.ST stat.TH