*3.2. Aggregational Gaussianity*

Aggregational Gaussianity is considered a stylized fact in traditional financial markets. In [50], the authors observed the evolution of distributions of the IBM stock returns by looking at different levels of granularity, e.g., 30 min, one day, one week and one month, finding evidence of Aggregational Gaussianity. Another study on this topic drawing the same conclusion is described in [51]. However, these authors used different stocks and a higher set of timescales from one day to one year, showing that this stylized fact is also true for stocks at coarser time resolutions.

We investigate whether Aggregational Gaussianity exists in our log-return time series using a set of four timescales: 30 min, 6 h, 12 h and 1 day. We observe this statistical aspect by implementing three experiments: Firstly, we construct the histogram as well as kernel density estimation (KDE) for each cryptocurrency time series. Secondly, we generate the Q-Q plot, which is a popular approach to test normality for a time series [52]. Lastly, we use the Lilliefors hypothesis test for normality [53]. We obtained the following findings: firstly, although the distributions of these cryptocurrency time series have a bell curve shape at all timescales considered, they are not (from the Q-Q plot and Lilliefors test) normally distributed; secondly, however, there appears to be evidence to say that Aggregational Gaussianity exists in all cryptocurrencies used in this present study from the Q-Q plots. This result is in line with existing findings in the cryptocurrency market such as [54,55].
