Let $X = (X_1,\ldots,X_n) \in \mathbb{R}^n$ be jointly Gaussian with mean $0$, covariance matrix: $Var(X_i) = 1$, $Cov(X_i, X_{i+1}) = -1/2$, and $Cov(X_i, X_j) = 0$ else.

Let $\zeta \in \mathbb{R}^n$ be a fixed vector, $\zeta_i > 0$ for all $i$. I want a tight lower bound on $P(X < \zeta)$, that is,

$$ P(X_1 \leq \zeta_1, \ldots, X_n \leq \zeta_n) \geq ? $$

In particular, I'm interested in the case where the $\zeta_i$'s are i.i.d $exp(1)$.

One rather loose bound when the $\zeta_i$'s are far apart is $$ P(X_1 \leq \zeta_1, \ldots, X_n \leq \zeta_n) \geq P(\max_iX_i \leq \min_j\zeta_j) $$ and from there one can lower bound this further with Sudakov.

Due to the negative correlation we can't apply Slepian's lemma directly (and the high negative correlation would probably give us a bad bound).

I've looked into the multivariate Mills ratio literature, but they seem to be concerned with bounding $$ P(X_i \geq \zeta_i) $$ for $\zeta_i > 0$

I've also tried writing down the integral and the inverse covariance matrix explicitly. (Since the covariance matrix $\Sigma$ in this case is a triangular circulant matrix, the inverse can be found in closed form, but it's not sparse and rather complicated).

Any ideas would be highly appreciated. This has applications in ranking and statistics.

Ngoc