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