You need to read the original paper by Getis and Ord that introduces the G statistics:
Getis A, Ord JK. 1992. The analysis of spatial association by use of distance statistics. Geographical Analysis 24:189-206.
See the paragraph that begins at the bottom of page 191.
To paraphrase, the sampling distribution of $G_i$ under the null hypothesis (of complete spatial randomness..."CSR") is asymptotically normal. The expected value is based on the number of points $j$ that are connected to point $i$ relative to the total number of points. When the distance is sufficiently small such that the expected value is small, normality is lost; likewise, normality is lost when all points $j$ are connected to $i$.
This suggests that it is OK to use the normal approximation when you have a lot of points and your expected values are not too close to zero or one. This also suggests that this decision is somewhat context dependent: it depends on the number of points and their configuration.
You can also use a permutation test to assess the significance of $G_i$ or $G_i^*$. If the P-values obtained based on the normal approximation are wildly different from those obtained by the permutation test, it might suggest that your expected values are violating the assumption of the normal approximation.