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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 

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