  The purpose of this paper is to show that the use of heavy-tailed
distributions in Financial problems is theoretically baseless and can lead to
significant misunderstandings. The reason for this the authors see in an
incorrect interpretation of the concept of the distributional tail. In
accordance with this, in applications it is necessary to use instead of the
distributional tail the "smearing" of its central part. keywords: heavy tailed
distributions, exponential tails, pre-limit theorems, Financial indexes
