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uniformly in 2 θ$_{pk}$, 2 θ$_{ps}$ ∈ Ω$_{εη}$, a.s.. 

A straightforward calculation, using formula (7), shows that 

uniformly in 2 θ$_{pk}$, 2 θ$_{ps}$ ∈ Ω$_{εη}$, a.s., where the matrix V ( k,s ) p has elements 

<img src='content_image/323718.jpg'>

<img src='content_image/323719.jpg'>

This 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.. 

<img src='content_image/323720.jpg'>

<img src='content_image/323721.jpg'>

<img src='content_image/323722.jpg'>

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 

$^{B$_{M,M}$}$′ where Φ ( x ; E $_{p}$γ$_{p}$ , V $_{p}$γ$_{p}$ ) denotes the Gaussian distribution function with mean 

<img src='content_image/323723.jpg'>

