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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?

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  • $\begingroup$ 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? $\endgroup$ – Lucas Farias May 20 '17 at 13:19
  • $\begingroup$ 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. $\endgroup$ – Ievgen May 20 '17 at 19:23
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As a comment mentioned, with such high excess kurtosis, arguing in favor of normality will be problematic.

The case has been made that stock returns can be adequately modeled using the Laplace distribution, (which has a high excess kurtosis) see

Kotz, S., Kozubowski, T., & Podgorski, K. (2001). The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer Science & Business Media.

...chapter 8, especially 8.4.2 and literature cited therein.

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