Marchenko-Pastur distribution function, with density where a$_{p}$ = ( 1 − √ c$_{p}$ ) 2 and b$_{p}$ = ( 1 + √ c$_{p}$ ) $^{2}$, and a mass of max ( 0 , 1 − c − 1 p ) at zero. As is well known, the difference between ˆ F λ p and F MP p weakly converges to zero a.s. as p, n$_{p}$ → c ∞ . Moreover, λ$_{p}$$_{1}$ a.s → $_{(1 +}$√$_{c}$$_{)}$ $^{2}$, and λ$_{pp}$ a.s → (1 − √$_{c}$$_{)}$ 2 if c > 1, and λ$_{pp}$ a.s → 0 if c ≤ 1 . Consider the Hilbert transform of F MP p , H MP p ( x ) = ∫ ( x − λ $_{)}$− $^{1}$d F MP p ( λ ) . That transform is well defined for real x outside the support of F MP p , that is, on the set R \ supp ( F MP p ) . Using (5), we get where the sign of the square root is chosen to be the sign of ( x − c$_{p}$ − 1). It is not hard to see that H MP p ( x ) is strictly decreasing on R \ supp ( F MP p ) . Thus, on H MP p ( R \ supp ( F MP p )) , we can define an inverse function K MP p , with values The so-called R -transform R MP p of F MP p takes the form For ε > 0 and η > 0 sufficiently small, consider the subset of R Ω$_{εη}$ =    $^{[}$$^{−}$$^{η}$$^{−}$$^{1}$$^{,}$$^{0)}$ ∪ ( 0 , 1 √$_{c}$$_{(}$$_{1+}$√$_{c}$$_{)}$ − ε ] for c ≥ 1 , [ − 1 √$_{c}$$_{(}$$_{1}$ − √$_{c}$$_{)}$ + ε, 0 ) ∪ ( 0 , 1 √$_{c}$$_{(}$$_{1+}$√$_{c}$$_{)}$ − ε ] for c < 1 . 9 R MP p ( x ) = K MP p ( x ) − 1 /x = 1 / (1 − c$_{p}$x ) . K MP p ( x ) = 1 x + 1 1 − c$_{p}$x , x ∈ H MP p ( R \ supp ( F MP p )) . (7) H MP p ( x ) = x + c$_{p}$ − 1 − √ ( x − c$_{p}$ − 1) 2 − 4 c$_{p}$ 2 c$_{p}$x , (6) f MP p ( x ) = 1 2 πc$_{p}$x √ ( b$_{p}$ − x ) ( x − a$_{p}$ ) , (5)