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Kolmogorov distance k ( µ, ν ) = sup$_{x}$ | F$_{µ}$ ( x ) − F$_{ν}$ ( x ) | , the Wasserstein distance w ( µ, ν ) = sup { ∣ ∫ f ( dµ − dν ) ∣ : ∣ f ( x ) − f ( y ) x y ∣ ≤ 1 , ∀ x ̸ = y } , and the Kantorovich distance γ ( µ, ν ) = ∫ | F$_{µ}$ ( x ) − F$_{ν}$ ( x ) | dx. As is well known (see, for example, ex- ercise 1 on p.425 of Dudley (2002)), w ( µ, ν ) = γ ( µ, ν ) . Therefore, we have d ( ˆ F λ p , F MP p ) ≤ w ( ˆ F λ p , F MP p ) = γ ( ˆ F λ p , F MP p ) ≤ k ( ˆ F λ p , F MP p ) ( diam ( ˆ F λ p ) + diam ( F MP p ) . As follows from Theorem 1.1 of G¨ otze and Tikhomirov (2011), there exists a 

constant τ > 0 such that ∑ ∞ p $_{=1}$Pr ( k ( ˆ F λ p , F MP p ) > εp − $^{1}$log τ p ) < ∞ for all ε > 0. Thus, k ( ˆ F λ p , F MP p ) = o ( p − $^{1}$log τ p ) a.s.. Since diam ( F MP p ) is O (1) and diam ( ˆ F λ p ) − diam ( F MP p ) → 0 a.s., the result follows. □ 

Corollary 8 Suppose that a sequence of functions { f$_{p}$ ( λ ) } is bounded Lipshitz on supp ( F MP p ) ∪ supp ( ˆ F λ p ) , uniformly over all sufficiently large p, a.s.. Then ∣ ∫ f$_{p}$ ( λ )d ˆ F λ p ( λ ) − F MP p ( λ )∣ ∣ = o ( p − 1 / 2 ) , a.s.. 

## 5.1 Proof of Proposition 2 

Let us denote the integral ∫ O ( p ) e p tr(Θ$_{p}$ Q ′ Λ$_{p}$ Q $^{)}$(d Q ) as I$_{p}$ (Θ$_{p}$ , Λ$_{p}$). As explained in Guionnet and Maida (2005, p.454), we can write 

where E Λ$_{p}$ denotes the expectation conditional on Λ$_{p}$ , and the p -dimensional vectors ( ˜ g $^{(1)}$, ..., ˜ g ( r ) ) are obtained from standard Gaussian p -dimensional vectors ( g $^{(1)}$, ..., g( r ) ) , independent from Λ$_{p}$, by a Schmidt orthogonalization procedure. More precisely, we have ˜ g ( j ) = ∑ j k $_{=1}$A$_{jk}$ g ( k $^{)}$, where A$_{jj}$ = 1 and j − 1 ( k ) ( t ) ( j ) ( t ) 

In the spirit of the proof of Guionnet and Maida’s (2005) Theorem 3, define 

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