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of this test is close to the envelope. 

## 5 Appendix 

<img src='content_image/299069.jpg'>

All convergence statements made below refer to the situation when n$_{p}$, p → c ∞ . We start with two auxiliary results. 

Lemma 7 Let d ( µ, ν ) be the Dudley distance between measures µ and ν defined over ( R , B ) : 

d ( µ, ν ) = sup {∣ ∣ ∫ f ( dµ − dν ) ∣ : f ( x ) ≤ 1 and ∣ ∣$^{f}$ ( x ) − f ( y ) x y ∣ ≤ 1 , ∀ x ̸ = y } . There exists a constant τ > 0 such that d ( ˆ F λ p , F MP p ) = o ( p − $^{1}$log τ p ) a.s.. 

Proof: Let us denote the cumulative distribution function corresponding to a measure µ as F$_{µ}$ ( x ) . Further, let us denote inf {| x$_{2}$ − x$_{1}$ | : supp ( µ ) ⊆ [ x$_{1}$, x$_{2}$ ] } as diam ( µ ) . Consider the following three distances between measures µ and ν : the 

