ИСТИНА

The role of multidimensional normal distribution in many applied statistical studies is well known. This is due, in particular, to the fact that it has the following good properties. The density of this distribution is written out explicitly, which makes it possible to use the maximum likelihood method to estimate the parameters. Its characteristic function is also easily written out explicitly, which makes it easy to write out its numerical characteristics, for example, the average of each component and the covariance matrix. Any linear combination of coordinates will be a distribution of the same type. This makes it easy to find the distribution of a single coordinate, the distribution of the sum of coordinates, the conditional distribution of one coordinate provided for the sum of all coordinates. But the tails of such a distribution decrease very quickly, which is often not done in many specific tasks. In our report, we consider a certain family of multidimensional distributions that has all the properties listed above, but, unlike a multidimensional normal distribution, the tails of such distributions decrease in a power-law manner, i.e. they are heavy. Some variant of such family has been considered earlier. But our definition is more general and we investigate it using a new method when these distributions are considered both as densities and as characteristic functions at the same time. In addition, their close connection with limit distributions for multidimensional geometric random sums is shown.