estimate ( ˜ B$_{n}$ , ˜ Σ$_{n}$ ) and Let be respectively an MM-estimate for the MLM and its initial estimate computed with the sample Z ∗ ∈ Z $_{m}$. Then by (2.6), (2.4) and (2.1) Moreover, since sup ρ$_{1}$ ( u ) = 1, we get Then there exists c < ∞ , that does not depend on Z $^{∗}$, such that, for at least [ n/ 2] observations of Z $^{∗}$, d 2 $_{i}$( ̂ B ∗ $_{n}$, ̂ Σ ∗ $_{n}$) < c . Now, since m < [ n/ 2] k$_{n}$ , at least k$_{n}$ + 1 of these observations are in Z , Now, since m < [ n/ 2] − k$_{n}$ , at least k$_{n}$ + 1 of these observations are in Z , and not in a hyperplane. Then the smallest eigenvalue of ̂ Σ ∗ $_{n}$, λ$_{q}$ ( ̂ Σ ∗ $_{n}$), is bounded below with a positive bound (for every x$_{i}$ ∈ R $^{p}$, the axis of the ellipsoid 37 { y : ( y − ̂ B ∗ n $^{′}$x$_{i}$ ) ′ ̂ Σ ∗− 1 n ( y − ̂ B ∗ n $^{′}$x$_{i}$ ) ≤ c } ∑ z$_{i}$ ∈ Z ∗ ρ$_{1}$ ( d$_{i}$ ( ̂ B ∗ $_{n}$, ̂ Σ ∗ $_{n}$) ) ≤ n 2 ρ$_{1}$ ( ∞ ) . 1 n ∑ z$_{i}$ ∈ Z ∗ ρ$_{1}$ ( d$_{i}$ ( ̂ B ∗ $_{n}$, ̂ Σ ∗ $_{n}$) ) ≤ 1 n ∑ z$_{i}$ ∈ Z ∗ ρ$_{1}$ ( d$_{i}$ ( ˜ B ∗ $_{n}$, ˜ Σ ∗ $_{n}$) ˆ σ ∗ n ) ≤ 1 n z$_{i}$ ∈ Z ∗ ρ$_{0}$ d$_{i}$ ( B ∗ $_{n}$, Σ ∗ $_{n}$) ˆ σ ∗ n = 0.5 . ( ̂ B ∗ $_{n}$, ̂ Σ ∗ $_{n}$) = ( ̂ B ∗ $_{n}$( Z $^{∗}$) , ̂ Σ ∗ $_{n}$( Z $^{∗}$)) and ( ˜ B ∗ $_{n}$, ˜ Σ ∗ $_{n}$) = ( ˜ B ∗ $_{n}$( Z $^{∗}$) , ˜ Σ ∗ $_{n}$( Z $^{∗}$)) m < min( nε ∗ $_{n}$( Z , ˜ B$_{n}$ , ˜ Σ$_{n}$ ) , [ n/ 2] − k$_{n}$ ) .