First, let us find the mean E $_{p}$γ$_{p}$ , and the variance V $_{p}$γ$_{p}$ of γ$_{p}$ under mea- sure ∏ r j =1 ∏ p i $_{=1}$d P ( j ) pi ( g ( j ) i . Note that V $_{p}$γ$_{p}$ = diag ( V $_{p}$γ (1 , 1) p , V $_{p}$γ (2 , 1) p ,..., V $_{p}$γ ( r,r ) p and E $_{p}$γ$_{p}$ = ( E $_{p}$γ (1 , 1) p , E $_{p}$γ (2 , 1) p ,..., E $_{p}$γ ( r,r ) p ) ′ . With probability one, for sufficiently large p, we have which, by Corollary 1, is o (1) uniformly in 2 θ$_{pk}$ ∈ Ω$_{εη}$, a.s.. That Corollary 1 can be applied here follows from the form of expression (7) for K MP p ( x ). Similarly, uniformly in 2 θ$_{pk}$, 2 θ$_{ps}$ ∈ Ω$_{εη}$, a.s.. Thus, Next, with probability one, for sufficiently large p we have uniformly in 2 θ$_{pk}$, 2 θ$_{ps}$ ∈ Ω$_{εη}$. Similarly, we have and 26 V $_{p}$γ ( k,s ) p 2 = 1 p p ∑ i =1 λ 2 $_{pi}$(1 + δ$_{ks}$ ) (1+2 θ$_{pk}$v$_{pk}$ − 2 θ$_{pk}$λ$_{pi}$ ) (1+2 θ$_{ps}$v$_{ps}$ − 2 θ$_{ps}$λ$_{pi}$ ) = 1+ δ$_{ks}$ 4 θ$_{pk}$θ$_{ps}$ ( 1+ K MP p (2 θ$_{ps}$ ) K MP p (2 θ$_{pk}$ ) H (2) p,ks − 2 θ$_{pk}$K MP p (2 θ$_{pk}$ ) − 2 θ$_{ps}$K MP p (2 θ$_{ps}$ ) + o (1) , V $_{p}$γ ( k,s ) p 1 = 1 + δ$_{ks}$ 4 θ$_{pk}$θ$_{ps}$ ˆ H (2) p,ks = 1 + δ$_{ks}$ 4 θ$_{pk}$θ$_{ps}$ H (2) p,ks + o (1) a.s. , V $_{p}$γ ( k,s ) p 1 = 1 p p ∑ i =1 1 + δ$_{ks}$ (1 + 2 θ$_{pk}$v$_{pk}$ − 2 θ$_{pk}$λ$_{pi}$ ) (1 + 2 θ$_{ps}$v$_{ps}$ − 2 θ$_{ps}$λ$_{pi}$ ) . Let ˆ H (2) p,ks = ∫ d ˆ F λ p ( λ ) ( K MP p ( 2 θ$_{pk}$ $^{)}$− λ )( K MP p (2 θ$_{ps}$ ) − λ ) and H (2) p,ks = ∫ d F MP p ( λ ) ( K MP p ( 2 θ$_{pk}$ $^{)}$− λ )( K MP p (2 θ$_{ps}$ ) − λ ) . Then, using Corollary 1, we get sup { 2 θ$_{pj}$ ∈ Ω$_{εη}$ ,j ≤ r }$^{E}$$^{$_{p}$γ$_{p}$}$ = o (1) a.s. . (29) E $_{p}$γ ( k,s ) p 2 = √ p δ$_{ks}$ 2 θ$_{pk}$ ∫ K MP p (2 θ$_{pk}$ ) K MP p (2 θ$_{pk}$ ) − λ d ( ˆ F λ p ( λ ) − F MP p ( λ ) = o (1) E $_{p}$γ ( k,s ) p 1 = √ pδ$_{ks}$ ( 1 p p ∑ i =1 1 (1 + 2 θ$_{pk}$v$_{pk}$ − 2 θ$_{pk}$λ$_{pi}$ ) − 1 ) = √ pδ$_{ks}$ ∫ (2 θ$_{pk}$ $_{)}$− 1 K MP p (2 θ$_{pk}$ ) − λ d ( ˆ F λ p ( λ ) − F MP p ( λ ) ) ,