Assets such as stocks usually display kurtosis in their logarithmic returns. However, their logarithmic returns in a time interval $n$ are the sum of smaller logarithmic returns in $1/n$ time intervals. In other words, their return distribution can always be decomposed as the sum of many other distributions.
Asset return time series, under the Efficient Market Hypothesis, are martingales: they possess no autocorrelation.
By the Central Limit Theorem, return distributions of such assets should be normal. Nonetheless, they exhibit kurtosis, which is absent in a normal-distributed variate since the gaussian distribution is zero-valued in all moments beyond the second. Indeed, the presence of kurtosis has been famously pointed as a flaw in Black-Scholes.
The question is: why is there kurtosis in asset returns?