Suppose $X_1,\dots,X_N \sim_{iid} \mathcal{N}(0,1)$ are iid normal, and let $K=N/2$.
Let $S$ denote the collection of all subsets of $\left\{1,\dots,N\right\}$ with $K$ elements.
For any $s\in S$ and $X \in \mathbb{R}^N$, let $g(X,s)=\sum_{i\in s}{X_i}$ denote the partial sum of the subvector corresponding to $s$
My goal is to find the appropriate sequence $C_K$ which satisfies the expression $$E \left[1\left\{\sum_{s\in S}{\exp(g(X,s)}) > C_K \right\}\right]=\alpha$$ under asymptotics where $K\rightarrow \infty$, for fixed $\alpha\in (0,1)$
It seems that this should amount to finding a convergence theorem for the empirical distribution of all partial sums: $\left\{g(X,s)\,:\,s\in S\right\}$ over probable realizations of $X$. Simulations suggest that such distributions are approximately-normal for most realizations of $X$