Checking normality when there is no independence

There is a realization $(\xi_1(\omega), \xi_2(\omega), \ldots \xi_n(\omega))$ of a random vector $$\boldsymbol{\xi}=(\xi_1, \xi_2, \ldots, \xi_n).$$ I'd like to check the following: is it true that all random variables $\xi_1, \xi_2, \ldots, \xi_n$ are normally (or at least approximately normally) distributed?

Our $\xi_1, \xi_2, \ldots, \xi_n$ are not independent, but they are uncorrelated or almost uncorrelated (I deal with successive returns of logs of stock prices).
Can we say that the random variables are not normal if the excess kurtosis of our "sample" is equal to 5.4? What tests of normality should be "at least approximately applicable" in such situation?

• If you're just concerned with the individual variables, couldn't you simply perform a normality test on them separately? Moreover, if there is excess kurtosis, isn't it immediate that this is not a Normal distribution? May 20, 2017 at 13:19
• I have only one realization. I'm not sure whether it's correct to apply the notion of the empirical kurtosis to such a "sample" of a vector consisting of random variables with possibly different distributions. May 20, 2017 at 19:23