Q: What kind of (time-series) periodic observations can we safely assume a normal distribution?
A: if each periodic observation is under the same, never-changing context
Example: suppose every day I pick a kid at random from my son’s school class and record the kid’s height. Since the inherent distribution of the class is normal, my periodic sample is kind of normal. However, kids grow fast, so there’s an uptrend in the time series. Context is changing. I won’t expect a real normal distribution in the time series data set.
In finance, majority of the important time-series data are price-related including vol and return. Prices change over time, sometimes on an uptrend, sometimes on a downtrend. Example: if I ask 100 analysts to forecast the upcoming IBM dividend, I could perhaps assume a Normal distribution, but not the time-series.
In conclusion, in a finance context my answer to the opening question is “seldom”.
I would even say that financial data is no natural science but behavior science. Seldom has an inherent Normal distribution. How about central limit theorem? It requires iid, usually not valid.