How can I find $N$ such that, for $g = \frac{1}{N} \sum_i^N g_i$,
$\left[ \mathbb E \frac{g}{\|g\|} \right]^\intercal \left[ \frac{\mathbb E g}{\|\mathbb E g\|}\right] \ge \kappa$,
where $g_1,\ldots,g_n$ are iid, and $\kappa$ is about 0.5? I don't know the distribution of the $g_i$, but I can draw samples of them fairly cheaply. Unfortunately I cannot explicitly form an estimate of the $cov(g_1)$ matrix because the dimension $d$ of $g$ is too large to store a $d \times d$ matrix.