we have that Then we can apply Lemma 18 and conclude that F$_{kj}$ is euclidean for the envelope with ξ$_{0}$ ∈ C such that θ ( ξ$_{0}$ ) = ( B$_{0}$ , σ 2 $_{0}$Σ$_{0}$ ) and R = 2 √ q ( p + q ) sup C ‖ ξ − ξ$_{0}$ ‖ . The proof of E$_{H}$$_{0}$F 2 kj < ∞ follows immediately using that | W ( u ) | ≤ M and expanding (A.52) as a sum of products, and bounding their respective means by means of (A6). ■ Before proving Theorem 6 we need to state Lemma 2.16 (page 1036) of Pakes and Pollard [18]. Lemma 20. Let F be a Euclidean class with envelope F such that ∫ F $^{2}$dP < ∞ . For each η > 0 and ε > 0 there exists a δ > 0 such that where [ δ ] represents the set of all pairs of functions in F with and ν$_{n}$ ( f ) = n − 1 / 2 n ∑ i =1 [ f ( ζ$_{i}$ ) − ∫ fdP ] , where ζ$_{1}$, ζ$_{2}$, . . . , ζ$_{n}$ are independent ob- servations sampled from the distribution P . The proof of Lemma 20 can be found in [18]. 54 ∫ ( f$_{1}$ − f$_{2}$ ) $^{2}$dP < δ2 lim sup P { sup [ δ ] | ν$_{n}$ ( f$_{1}$ ) − ν$_{n}$ ( f$_{2}$ ) | > η } < ε, | φ$_{kj}$ ( z ; θ ( ξ )) − φ$_{kj}$ ( z ; θ ( ξ $^{∗}$)) | ≤ ϕ$_{kj}$ ( z ) ‖ ξ − ξ $^{∗}$‖ . F$_{kj}$ ( z ) = | φ$_{kj}$ ( z , θ ( ξ$_{0}$ )) | + Rϕ$_{kj}$ ( z ) ,