From (6), H MP p ( R \ supp ( F MP p )) = ( −∞ , 0) ∪ ( 0 , 1 √ c$_{p}$ ( 1+ √ c$_{p}$ ) ) ∪ ( 1 √ c$_{p}$ $_{(}$√ c$_{p}$ − 1 ) , ∞ ) when c$_{p}$ > 1, − 1 √ c$_{p}$ ( 1 − √ c$_{p}$ ) , 0 ) ∪ ( 0 , 1 √ c$_{p}$ ( 1+ √ c$_{p}$ ) ) when c$_{p}$ < 1, and ( −∞ , 0) ∪ (0 , 1 / 2) when c$_{p}$ = 1. Therefore, Ω$_{εη}$ ⊂ H MP p ( R \ supp ( F MP p )) with probability approaching one as n$_{p}$, p → c ∞ . Proposition 2 Let { Θ$_{p}$ } be a sequence of random p × p diagonal matrices diag ( θ$_{p}$$_{1}$, ..., θ$_{pr}$, 0 , ..., 0) , where θ$_{pj}$ = 0 , j = 1 , ..., r. Further, let v$_{pj}$ = R MP p (2 θ$_{pj}$ ) , where R MP p ( x ) = 1 / (1 − c$_{p}$x ) is the R -transform of the Marchenko-Pastur distri- bution F MP p . Assume that, for some ε > 0 and η > 0 , 2 θ$_{pj}$ ∈ Ω$_{ε,η}$ with probability approaching one as n$_{p}$, p → c ∞ . Then, where o (1) is uniform over all sequences { Θ$_{p}$ } satisfying the assumption. This proposition extends Theorem 3 of Guionnet and Maida (2005) to cases when rank (Θ$_{p}$) > 1 , θ$_{pj}$ depends on p, and Λ$_{p}$ is random. When r = 1 , θ$_{p}$$_{1}$ = θ > 0 and v$_{p}$$_{1}$ = v are fixed, it is straightforward to verify that √ 1 − 4 θ $^{2}$v$^{2}$c$_{p}$ = √$_{4}$$_{θ}$ 2 $_{/}$√$_{Z,}$ where Z = ∫ ( K MP p (2 θ ) − λ ) − $^{2}$d F MP p ( λ ) . In Guionnet and Maida’s (2005) Theorem 3, the expression √$_{4}$$_{θ}$ 2 $_{/}$√$_{Z}$ should have been used instead of √ Z − 4 θ 2 $_{/θ}$√$_{Z,}$ which is a typo. 1 h Setting r = 1 and θ$_{p}$$_{1}$ = 2 c$_{p}$ 1+ h in Proposition 2 and using formula (3) from Proposition 1 gives us an expression for L$_{p}$ ( h ; λ$_{p}$ ) which is an equivalent of formula (4.1) in Theorem 7 of OMH. Theorem 3 below uses Proposition 2 to generalize Theorem 7 of OMH to the multispiked case r > 1. 10