Cov ( η, ζ ) = − 1 4 π 2 ∮ ∮ z ( m$_{1}$ ) ( z ( m$_{2}$ )) 2 ( m$_{1}$ − m$_{2}$ ) 2 d m$_{1}$ d m$_{2}$, and (59) Var ( ζ ) = − 1 4 π 2 ∮ ∮ ( z ( m$_{1}$ )) $^{2}$( z ( m$_{2}$ )) 2 ( m$_{1}$ − m$_{2}$ ) 2 d m$_{1}$ d m$_{2}$, (60) z ( m ) = − 1 m + c 1 + m (61) ∮ ln (¯ z$_{j}$$_{0}$ − z ( m$_{1}$ )) ( m$_{1}$ m$_{2}$ ) 2 d m$_{1}$ = 2 π i ( − 1 m$_{2}$ + 1 m$_{2}$ + h$_{j}$ ( h$_{j}$ + c ) − 1 ) . (62) Cov ( ξ$_{j}$, ζ ) = 2 π i 4 π 2 ∮ ( z ( m$_{2}$ )) 2 ( 1 m$_{2}$ − 1 m$_{2}$ − x$_{j}$ ) d m$_{2}$ = 2 π i 4 π 2 ( − 1 m$_{2}$ + c 1 + m$_{2}$ ) 2 ( 1 m$_{2}$ − 1 m$_{2}$ − x$_{j}$ ) d m$_{2}$ = − h$_{j}$ ( h$_{j}$ + 2 + 2 c ) , where the last equality follows from Cauchy’s residue theorem and the fact that Denoting − h$_{j}$ ( h$_{j}$ + c ) − 1 as x$_{j}$, we get from (62) and (58) Assuming without loss of generality that m$_{1}$ -contour lies inside the m$_{2}$ -contour, from (A64) in the Supplementary appendix of OMH, we have and the contours of integration over m$_{1}$ and m$_{2}$ in (58-60) are obtained from the contours of integration over z$_{1}$ and z$_{2}$ in (54-56) by transformation m ( z ) . Recall that by assumption the contours over z$_{1}$ and z$_{2}$ intersect the real line to the left of zero and in between the upper boundary of the support of the Marchenko-Pastur $_{distribution, (1 +}$√$_{c}$$_{)}$ $^{2}$, and ¯ z$_{j}$$_{0}$. Therefore, as can be shown using the definition (57) of m ( z ), the m$_{1}$ -contour and m$_{2}$ -contour are clockwise oriented and intersect the real line in between − $_{(1 +}$√$_{c}$$_{)}$ − 1 and m (¯ z$_{j}$$_{0}$ ) = − h$_{j}$ ( h$_{j}$ + c ) − 1 and to the right of zero. In particular, both contours enclose 0 and − h$_{j}$ ( h$_{j}$ + c ) − $^{1}$, but not − 1 and − (1 + h$_{j}$ ) − $^{1}$. where 35