0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.3006 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.6683 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.9520 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.9995 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 of this test is close to the envelope. 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 22 5 Appendix Figure 7: Profiles of the asymptotic power of the Ledoit-Wolf and Cai-Ma tests (solid lines) and the CLR test (dashed lines, for c = 0 . 5) relative to the asymptotic power envelope β$_{λ}$ (dotted lines) for different values of h$_{1}$$_{/}$√$_{c}$ under the alternative; α = 0 . 05 .