and By Lemma 12 we have for all B ∈ R p × q and Γ ∈ S q with | Γ | = 1 such that Γ = I$_{q}$ . By Lemma 9, (A.47) and the Lebesgue dominated convergence Theorem, using a standard compactness argument we can find σ$_{1}$ > σ$_{0}$ , γ > 0 and a finite number of sets, C $_{1}$, . . . , C $_{s}$, such that and By (A.49) we have Then by (A.48) and the Law of Large Numbers we get (A.44). (A.45) is proved similarly to (A.44) and (A.46) is a consequence of Lemma 17. ■ Next we will give some definitions and lemmas that will be necessary to prove the asymptotic normality of MM-estimates ̂ B$_{n}$ . 50 lim n −→∞ inf ( B , Γ ) ∈ C$_{1}$ ( ε 1 ) 1 n n ∑ i =1 ρ$_{1}$ ( d$_{i}$ ( B , Γ ) σ$_{1}$ ) ≥ inf 1 ≤ j ≤ s lim n −→∞ 1 n n ∑ i =1 inf ( B , Γ ) ∈ C j ρ$_{1}$ ( d$_{i}$ ( B , Γ ) σ$_{1}$ ) . s ⋃ j =1 C j ⊃ C $_{1}$( ε$_{1}$ ) . (A.49) E$_{H}$$_{0}$ inf ( B , Γ ) ∈ C j ρ$_{1}$ ( d ( B , Γ ) σ$_{1}$ ) > E$_{F}$$_{0}$ρ$_{1}$ ( ( u $^{′}$u ) 1 / 2 σ$_{0}$ ) + γ (A.48) Eρ$_{1}$ ( d ( B , Γ ) σ$_{0}$ ) > Eρ$_{1}$ ( ( u $^{′}$u ) 1 / 2 σ$_{0}$ ) (A.47) lim n −→∞ 1 n n ∑ i =1 ρ$_{1}$ ( d$_{i}$ ( ˜ B$_{n}$ , ˜ Σ$_{n}$ ) ˆ σ$_{n}$ ) = E$_{F}$$_{0}$ρ$_{1}$ ( ( u $^{′}$u ) 1 / 2 σ$_{0}$ ) a.s.. (A.46)