I'm testing a short call option strategy and found, as expected, non-normal return distributions. It is known that option returns are not normally distributed (i.e., also the population). I take the options from a population of 5 million different options and have in the end about 6000 different options options (i.e., my sample is N>30). I am woundering whether the central limit theorem holds for non-normal populations and non-normal samples? In the end, I would like to perform one-sample one-sided t-tests for the mean.