content_image/311620.jpg content_image/311621.jpg content_image/311619.jpg content_image/311617.jpg content_image/311618.jpg
of vectors with odd coordinates in ( − M, M ) and even coordinates in ( − M ′ , M ′ ). 

Let 

<img src='content_image/311620.jpg'>

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 

<img src='content_image/311621.jpg'>

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, 

<img src='content_image/311619.jpg'>

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 

<img src='content_image/311617.jpg'>

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 $^{2}$+ $^{r}$. 

<img src='content_image/311618.jpg'>

