and By the Lebesgue dominated convergence Theorem we can take 0 < η < δ such that Then using the Law of Large Numbers we obtain and get (A.42). A similar procedure is performed to prove (A.43). ■ Proof of Theorem 5: Consider According to the Lemmas 14 and 16 and (2.6), it would be enough to show that given ε$_{1}$ > 0, ε$_{2}$ > 0 and L$_{1}$ and L$_{2}$ arbitrarily large, there exist γ > 0 and σ$_{1}$ > σ$_{0}$ such that 49 lim$_{n}$$_{−→∞}$ inf ( B , Γ ) ∈ C$_{1}$ ( ε 1 ) 1 n n ∑ i =1 ρ$_{1}$ ( d$_{i}$ ( B , Γ ) σ$_{1}$ ) ≥ E$_{F}$$_{0}$ρ$_{1}$ ( ( u $^{′}$u ) 1 / 2 σ$_{0}$ ) + γ a.s., (A.44) lim$_{n}$$_{−→∞}$ inf ( B , Γ ) ∈ C$_{2}$ ( ε 2 ) 1 n n ∑ i =1 ρ$_{1}$ ( d$_{i}$ ( B , Γ ) σ$_{1}$ ) ≥ E$_{F}$$_{0}$ρ$_{1}$ ( ( u $^{′}$u ) 1 / 2 σ$_{0}$ ) + γ a.s. (A.45) lim$_{n}$$_{−→∞}$ inf ‖| ( A , V ) − ( A$_{0}$ , V$_{0}$ ) ‖|≤ $_{η}$(1 /n ) n ∑ i =1 g ( z$_{i}$ , ( A , V )) ≥ E$_{Q}$ g ( z , ( A$_{0}$ , V$_{0}$ )) − ε (A.43) E ( sup ‖| ( A , V ) − ( A$_{0}$ , V$_{0}$ ) ‖|≤ η g ( z , ( A , V ))) ≤ E$_{Q}$ g ( z , ( A$_{0}$ , V$_{0}$ )) + ε. lim n →∞ 1 n n ∑ i =1 sup ‖| ( A , V ) − ( A$_{0}$ , V$_{0}$ ) ‖|≤ η g ( z$_{i}$ , ( A , V ))= E ( sup ‖| ( A , V ) − ( A$_{0}$ , V$_{0}$ ) ‖|≤ η g ( z , ( A , V ))) C ( δ, L$_{1}$, L$_{2}$ ) = { ( B , Γ ) ∈ R p × q × S q : δ ≤ ‖ Γ ‖ ≤ L$_{1}$, | Γ | = 1 and ‖ B ‖ 2 ≤ L$_{2}$ } , C $_{1}$( ε ) = { ( B , Γ ) ∈ C ( δ, L$_{1}$, L$_{2}$ ) : ‖ B ‖ 2 ≥ ε } and C $_{2}$( ε ) = { ( B , Γ ) ∈ C ( δ, L$_{1}$, L$_{2}$ ) : ‖ Γ − I$_{q}$ ‖ ≥ ε } .