This paper examines "fat tails puzzle" on the financial markets. Ignoring the rate of convergence in the framework of the Central Limit Theorem (CLT) provides the "fat tail" uncertainty. In this paper, we provide empirical results obtained for the “fat tails puzzle” using innovative method of Yuri Gabovich. The approach builds up the so-called G-bounds based on the estimates of rate of convergence to the asymptotic normal distribution. Constructed G-bounds evaluate risk on the financial markets more carefully than models based on Gaussian distribution. This statement was tested on the 24 financial markets exploring their stock indexes. Besides this there have been tested Weak-Form Market Efficiency Hypothesis for investigated markets. As a result, we found out the negative correlation between the propensity of the stock market to be weak effective and the thickness of the left tail of the sample profitability density function. Therefore, the closer the risk of losses on the stock market to the corresponding risk of loss for a normal distribution, there is the higher probability that the market is weak effective. For non - effective markets, the probability of large losses is much higher than for a weak effective.