of vectors with odd coordinates in ( − M, M ) and even coordinates in ( − M ′ , M ′ ). Let where 1 {·} denotes the indicator function. Below, we establish the asymptotic behavior of I M,M ′ p (Θ$_{p}$ , Λ$_{p}$) as first p, and then M and M ′ , diverge to infinity. We then show that the asymptotics of I M,M ′ p (Θ$_{p}$ , Λ$_{p}$) and I$_{p}$ (Θ$_{p}$ , Λ$_{p}$) coincide. Consider infinite arrays { P ( j ) pi , p = 1 , 2 , ... ; i = 1 , ..., p } , j = 1 , ..., r, of random centered Gaussian measures Since v$_{pj}$ = R MP p (2 θ$_{pj}$ ) = 1 / (1 − 2 θ$_{pj}$c$_{p}$ ) and 2 θ$_{pj}$ ∈ Ω$_{εη}$, there exists ˆ ε > 0 such that, for sufficiently large p, Recall that λ$_{pp}$ ≥ 0 , and λ$_{p}$$_{1}$ → (1 + √$_{c}$$_{)}$ 2 a.s.. Therefore, still a.s., for sufficiently large p, v$_{pj}$ + 1 2 θ$_{pj}$ > λ$_{p}$$_{1}$ when θ$_{pj}$ > 0 and v$_{pj}$ + 1 2 θ$_{pj}$ < λ$_{pp}$ when θ$_{pj}$ < 0 . Hence, the measures P ( j ) pi are a.s. well defined for sufficiently large p . Whenever P ( j ) pi is not well defined, we re-define it arbitrarily. We have where We now show that, under ∏ r j =1 ∏ p i =1 d P ( j ) pi g ( j ) i ) , γ$_{p}$ a.s. converges in distribution to a centered r 2 + r -dimensional Gaussian vector, so that J M,M ′ p is asymptotically equivalent to an integral with respect to a Gaussian measure on R r $^{2}$+ $^{r}$. I M,M ′ p (Θ$_{p}$ , Λ$_{p}$) = e p ∑ r j $_{=1}$[ θ$_{pj}$v$_{pj}$ − 1 2 p ∑ p i $_{=1}$ln(1+2 θ$_{pj}$v$_{pj}$ − 2 θ$_{pj}$λ$_{pi}$ ) ] J M,M ′ p , (27) J M,M ′ p = ∫ 1 { B$_{M,M}$$_{′}$ } f$_{p,θ}$ ( γ$_{p}$ ) r ∏ j =1 p ∏ i =1 d P ( j ) pi ( g ( j ) i ) . (28) v$_{pj}$ + 1 2 θ$_{pj}$ > ( 1 + √$_{c}$ ) 2 + ˆ ε when θ$_{pj}$ > 0 and v$_{pj}$ + 1 2 θ$_{pj}$ < − ˆ ε when θ$_{pj}$ < 0 . I M,M ′ p (Θ$_{p}$ , Λ$_{p}$)= ∫ 1 { B$_{M,M}$$_{′}$ } f$_{p,θ}$ ( γ$_{p}$ ) e p ∑ r j $_{=1}$θ$_{pj}$ ( v$_{pj}$ +ˆ γ ( j,j ) p − v$_{pj}$γ ( j,j ) p ) r ∏ j =1 p ∏ i =1 d P ( g ( j ) i ) , d P ( j ) pi ( x ) = √ 1 + 2 θ$_{pj}$v$_{pj}$ − 2 θ$_{pj}$λ$_{pi}$ 2 π $_{e}$− 1 $_{2}$(1+2 θ$_{pj}$v$_{pj}$ − 2 θ$_{pj}$λ$_{pi}$ ) x $^{2}$d x. 25