C ov$_{p}$ ( γ ( k,s ) p 1 , γ ( k,s ) p 2 = 1 p p ∑ i =1 λ$_{pi}$ (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$_{ps}$ ( K MP p (2 θ$_{ps}$ ) H (2) p,ks − 2 θ$_{pk}$ ) + o (1) , H (2) p,ks = ( 1 4 θ$_{pk}$θ$_{ps}$ − c$_{p}$v$_{pk}$v$_{sk}$ ) − 1 , and V $_{p}$γ ( k,s ) p = V ( k,s ) p + o (1) , (30) V ( k,s ) p, 11 = (1+ δ$_{ks}$ ) (1 − 4 θ$_{pk}$v$_{pk}$θ$_{ps}$v$_{sk}$c$_{p}$ $_{)}$− 1 , (31) V ( k,s ) p, 12 = V ( k,s ) p, 21 =(1+ δ$_{ks}$ ) v$_{pk}$v$_{sk}$ (1 − 4 θ$_{pk}$v$_{pk}$θ$_{ps}$v$_{sk}$c$_{p}$ $_{)}$− 1 , and (32) V ( k,s ) p, 22 = (1+ δ$_{ks}$ ) [ c$_{p}$v$_{pk}$v$_{sk}$ + v 2 $_{pk}$v 2 $_{sk}$(1 − 4 θ$_{pk}$v$_{pk}$θ$_{ps}$v$_{sk}$c$_{p}$ $_{)}$− 1 ] . (33) det ( V ( k,s ) p ) = r ∏ k ≥ s (1+ δ$_{ks}$ ) 2 c$_{p}$v$_{pk}$v$_{sk}$ (1 − 4 θ$_{pk}$v$_{pk}$θ$_{ps}$v$_{sk}$c$_{p}$ $_{)}$− 1 , (34) J M,M ′ p = ∫ $^{B$_{M,M}$}$′ f$_{p,θ}$ ( x ) dΦ ( x ; E $_{p}$γ$_{p}$ , V $_{p}$γ$_{p}$ ) + o$_{M,M}$$_{′}$ (1) , (35) where Φ ( x ; E $_{p}$γ$_{p}$ , V $_{p}$γ$_{p}$ ) denotes the Gaussian distribution function with mean By construction, γ$_{p}$ is a sum of p independent random vectors having uniformly bounded third and fourth absolute moments under measure ∏ r j =1 ∏ p i $_{=1}$d P ( j ) pi ( g ( j ) i ) . Therefore, a central limit theorem applies. Moreover, since the function f$_{p,θ}$ ( γ$_{p}$ is Lipshitz over B$_{M,M}$$_{′}$, uniformly in { 2 θ$_{pj}$ ∈ Ω$_{εη}$ , j ≤ r } , Theorem 13.3 of Bhat- tacharya and Rao (1976), which describes the accuracy of the Gaussian approx- imations to integrals of the form (28) in terms of the oscillation measures of the integrand, implies that which is bounded away from zero and infinity for sufficiently large p , uniformly over { 2 θ$_{pj}$ ∈ Ω$_{εη}$ , j ≤ r } , a.s.. This implies that uniformly in 2 θ$_{pk}$, 2 θ$_{ps}$ ∈ Ω$_{εη}$, a.s., where the matrix V ( k,s ) p has elements A straightforward calculation, using formula (7), shows that uniformly in 2 θ$_{pk}$, 2 θ$_{ps}$ ∈ Ω$_{εη}$, a.s.. 27