We shall first need an auxiliary result, based on a proof in (Bianco and Yohai, 1996). Lemma 2 Assume model (2) with conditions A-B-C-D-E and A compact . Then ∥ ˆ β$_{n}$ ∥ is ultimately bounded with probability one. Proof of the Lemma: Put It follows from (C) that λ ( α, β ) attains its minimum only when β $^{′}$h ( x , α ) = β ′ $_{0}$h ( x , α$_{0}$ ) a.s. and by (9) this happens when ( α, β ) = ( α$_{0}$, β$_{0}$ ) . Therefore Let Γ = { γ ∈ B : ‖ γ ‖ = 1 } . Then we may write β = tγ with t = ‖ β ‖ ∈ R$_{+}$ and γ ∈ Γ . We divide the proof into two cases. Case I: bounded ρ : Assume that S = sup$_{u}$ ρ ( u ) < ∞ . To simplify notation it will be assumed without loss of generality that S = 1 . For each ( α, γ ) ∈ A × Γ we have where δ is defined in (10). Let Since (10) implies that P ( | γ $^{′}$h ( x , α ) | > 0) ≥ 1 − δ for γ ∈ Γ , then for each ( α, γ ) ∈ A × Γ there are positive a, b such that 7 P ( | y | ≤ a, | γ $^{′}$h ( x , α ) | ≥ b ) ≥ 1 − δ − ε. (12) ξ = 1 − δ − λ$_{0}$ > 0 , ε = ξ 4 < 1 − δ 4 . lim t $_{→∞}$E ρ ( y − tγ $^{′}$h ( x , α )) ≥ 1 − δ > λ$_{0}$, ( α, β ) = ( α$_{0}$, β$_{0}$ ) ⇒ λ ( α, β ) > λ ( α$_{0}$, β$_{0}$ ) = λ$_{0}$. (11) λ ( α, β ) = E ρ ( y − β $^{′}$h ( x , α )) .