stein (2004). The same theorem implies that where ¯ z$_{j}$$_{0}$ = lim z$_{j}$$_{0}$ as n, p → ∞ , and where with m ( z ) given by (3.6) of OMH, where c$_{p}$ is replaced by c. That is, where the branch of the square root is chosen so that the real and the imaginary parts of √ ( z − c − 1) 2 − 4 c have the same signs as the real and the imaginary parts of z − c − 1, respectively. The contours of integration in (54)-(56) are closed, oriented counterclockwise, enclose zero and the support of the Marchenko-Pastur distribution with parameter c , and do not enclose ¯ z$_{j}$$_{0}$ . The above expressions can be simplified. Use formula 1.16 of Bai and Silverstein (2004), to get 34 Cov ( ξ$_{j}$, ζ ) = − 1 4 π 2 ∮ ∮ ln (¯ z$_{j}$$_{0}$ − z ( m$_{1}$ )) ( z ( m$_{2}$ )) 2 ( m$_{1}$ − m$_{2}$ ) 2 d m$_{1}$ d m$_{2}$, (58) m ( z ) = − z + c − 1 + √ ( z − c − 1) 2 − 4 c 2 z , (57) m ( z ) = − (1 − c ) z − 1 + cm ( z ) Var ( ζ ) = − 1 4 π 2 ∮ ∮ z 2 $_{1}$z 2 ( m ( z$_{1}$ ) − m ( z$_{2}$ )) 2 d m ( z$_{1}$ ) d z$_{1}$ d m ( z$_{2}$ ) d z$_{2}$ d z$_{1}$ d z$_{2}$, (56) Cov ( η, ζ ) = − 1 4 π 2 ∮ ∮ z 2 $_{2}$z$_{1}$ ( m ( z$_{1}$ ) − m ( z$_{2}$ )) 2 d m ( z$_{1}$ ) d z$_{1}$ d m ( z$_{2}$ ) d z$_{2}$ d z$_{1}$ d z$_{2}$, (55) Cov ( ξ$_{j}$, ζ ) = − 1 4 π 2 ∮ ∮ z 2 ln (¯ z$_{j}$$_{0}$ − z$_{1}$ ) ( m ( z$_{1}$ ) − m ( z$_{2}$ )) 2 d m ( z$_{1}$ ) d z$_{1}$ d m ( z$_{2}$ ) d z$_{2}$ d z$_{1}$ d z$_{2}$, (54)