dard normal random variable. Further, as long as 2 θ$_{pj}$ ∈ Ω$_{εη}$ for j ≤ r, ˜ λ$_{pi}$ consid- ered as a function of λ$_{i}$ is continuous on λ$_{i}$ ∈ supp ˆ F λ p for sufficiently large p, a.s.. Hence, the empirical distribution of ˜ λ$_{i}$ converges. Moreover, ˜ λ$_{max}$ = max$_{i}$$_{=1}$$_{,...,p}$( ˜ λ$_{pi}$ ) and ˜ λ$_{min}$ = min$_{i}$$_{=1}$$_{,...,p}$( ˜ λ$_{pi}$ ) a.s. converge to finite real numbers. Now, for τ such that | τ | < √ p/ (2 ˜ λ$_{max}$ ) , we have for sufficiently large p, a.s.. Using this inequality, we get, for sufficiently large p and any positive t such that β$_{1}$r $^{2}$M + t < √ p/ (2 ˜ λ$_{max}$ ) , ∫ e τγ ( k,s ) p 2 r ∏ j =1 p ∏ i =1 d P ( j ) pi ( g ( j ) i ) = E e τ √ p 1 p ∑ p i =1 ˜ λ$_{pi}$ ˜ g ( k ) i ˜ g ( s ) i = p ∏ i =1 E e τ 1 √ $_{p}$˜ λ$_{pi}$ ˜ g ( k ) i ˜ g ( s ) i = p ∏ i =1 ( 1 − τ $_{2}$˜ λ 2 pi p ) − 1 / 2 ≤ e 2 ˜ λ 2 $_{max}$τ 2 ∫ γ ( k,s ) p 2 ≥ $^{M}$′ e β$_{1}$r $^{2}$Mγ ( k,s ) p 2 r ∏ j =1 p ∏ i =1 d P ( j ) pi ( g ( j ) i ≤ ∫ e β$_{1}$r $^{2}$Mγ ( k,s ) p 2 + t ( γ ( k,s ) p 2 − M ′ ) r ∏ j =1 p ∏ i =1 d P ( j ) pi ( g ( j ) i = $_{e}$− tM ′ ∫ e ( β$_{1}$r $^{2}$M + t ) γ ( k,s ) p 2 r j =1 p i =1 d P ( j ) pi ( g ( j ) i ) ≤ $_{e}$− tM $^{′}$e 2 ˜ λ 2 max ( β$_{1}$r $^{2}$M + t ) 2 . ∫ γ ( k,s ) p 2 ≥ $^{M}$′ e β$_{1}$r $^{2}$Mγ ( k,s ) p 2 r ∏ j =1 p ∏ i =1 d P ( j ) pi ( g ( j ) i ≤ $_{e}$− ( M ′ ) 2 8 ˜ λ 2 max + β$_{1}$r $^{2}$MM $^{′}$. ∫$_{∣}$ ∣ γ ( k,s ) p 2 ∣ ≥ $^{M}$′ e β$_{1}$r $^{2}$M ∣ γ ( k,s ) p 2 ∣ r ∏ j =1 p ∏ i =1 d P ( j ) pi ( g ( j ) i ≤ 2 $_{e}$− ( M ′ ) 2 8 ˜ λ 2 max + β$_{1}$r $^{2}$MM $^{′}$. ∫$_{∣}$ ∣ γ ( k,k ) p 2 ∣ ≥ $^{M}$′ e β$_{1}$r $^{2}$M ∣ γ ( k,k ) p 2 ∣ r ∏ j =1 p ∏ i =1 d P ( j ) pi ( g ( j ) i ≤ 4 $_{e}$− ( M ′ ) 2 16 ˜ λ 2 max + β$_{1}$r $^{2}$MM $^{′}$. J M,M ′ , ∞ p ≤ 4 r 2 $_{e}$− ( M ′ ) 2 16 ˜ λ 2 max + β$_{1}$r $^{2}$MM $^{′}$. (46) and thus, for sufficiently large p, When k = s, following a similar line of arguments, we obtain Replacing γ ( k,s ) p 2 by − γ ( k,s ) p 2 in the above derivations and combining the result with the above inequality, we get Setting t = M ′ 4 ˜ λ 2 max − β$_{1}$r $^{2}$M (here we assume that M and M ′ are such that t satisfies the above requirements), we get 31