matrices that is not a finite perturbation of the identity matrix. (See also [@El], in which the largest eigenvalue distribution was also derived for ensembles whose covariance matrix is not a finite perturbation of the identity matrix. However, in [@El], the parameters $c=\frac{N}{M}$, $\beta=\frac{N_1}{N}$ and the eigenvalues of the covariance matrix have to satisfy a condition which is not true in our case.)

In this paper, we use an important result by Baik, Ben-Arous and Péché [@Baik95] which shows that the correlation functions of the eigenvalues can be expressed in terms of a Fredholm determinant with kernel $K_{M,N}(x,y)$. In [@BK1] and [@DF], the authors have expressed this kernel in terms of multiple orthogonal polynomials (See Section \[se:MOP\] for details) and have shown that the $m$-point correlation function for the Wishart distribution (\[eq:wishart\]) is given by $$\label{eq:mpoint}
\mathcal{R}_{m}^{(M,N)}(y_1,\ldots,y_m)=\det\left(K_{M,N}(y_j,y_k)\right)_{1\leq
j,k\leq m}$$ where $\mathcal{R}_{m}^{(M,N)}(y_1,\ldots,y_m)$ is the $m$-point correlation function $$\label{eq:corre}
\mathcal{R}_m^{(M,N)}(y_1,\ldots,y_m)=\frac{N!}{(N-m)!}\int_{\mathbb{R}_+}\cdots\int_{\mathbb{R}_+}
\mathcal{P}(y)dy_{m+1}\ldots dy_N.$$ By computing the asymptotics of the correlation kernel in (\[eq:mpoint\]) through the asymptotics of multiple Laguerre polynomials, we have proven the universality of the correlation function when $M$, $N$ and $N_1\rightarrow\infty$.

\[thm:main2\] Suppose $\Delta$ in Theorem \[thm:main1\] is negative. Let $\rho(z)$ be the density function of $F$ in Theorem \[thm:side1\]. 