This is a problem I need to address for a certain computation in my research.

Let $Y_1,\dots,Y_n$ be a sequence of i.i.d. standard normal variables; and let $I\subset[0,+\infty)$ be an interval. In my application, if it helps one can think of $I=[a_n,b_n]$ where both $a_n,b_n$ are positive sequences that are both $o_n(1)$ that is $a_n,b_n\to 0$ as $n\to+\infty$.

Evaluate the following probability:
$$
\mathbb{P}\left(\bigcap_{1\leq i\leq n}\left\{\sum_{1\leq j\leq n,j\ne i}Y_iY_j\le 0 \right\}\cap \left\{\sum_{1\le j\le n}Y_j \in I\right\}\right).
$$
In particular,
**(a)** Is there a good way to compute this probability?
**(b)** How does it behave as a function of the interval $I$, equivalently, as a function of $a_n,b_n$.

If we denote the sum by $S$, the condition $\sum_{1\le j\le n,j\ne i}Y_iY_j\le 0$ is equivalent to having $Y_i^2\ge SY_i$, which, on top of $S\ge 0$ (recall $I\subset[0,\infty)$) implies either $Y_i\le 0$ or $Y_i\ge S$. In particular, as $I$ gets larger, $\mathbb{P}(S\in I)$ gets larger, whereas $\mathbb{P}(Y_i\in [0,S]^c)$ gets smaller, which obviously means the size of $I$ brings a compromise.

I could not see a good way of computing this, and appreciate any help.

evaluatethe probability (i.e. give a closed-form expression), or you would be satisfied with lower or upper bounds for that quantity? $\endgroup$ – Mateusz Kwaśnicki Jul 11 '20 at 23:03