Section 2

From a Formalization of Uncertainities to Probabilistic Modeling, Reasonable Control and Artificial Intelligence

**Chapter 7**

**Abstract**

sample quantiles

**1. Introduction**

**167**

Sample Sizes

From Asymptotic Normality

Theorems for Random Sums

and Statistics with Random

*Victor Korolev and Alexander Zeifman*

to Heavy-Tailedness via Limit

This chapter contains a possible explanation of the emergence of heavy-tailed distributions observed in practice instead of the expected normal laws. The bases for this explanation are limit theorems for random sums and statistics constructed from samples with random sizes. As examples of the application of general theorems, conditions are presented for the convergence of the distributions of random sums of independent random vectors *with finite covariance matrices* to multivariate elliptically contoured stable and Linnik distributions. Also, conditions are presented

for the convergence of the distributions of asymptotically normal (in the traditional sense) statistics to multivariate Student distributions. The joint

**Keywords:** random sum, random sample size, multivariate normal mixtures, heavy-tailed distributions, multivariate stable distribution, multivariate Linnik distribution, Mittag-Leffler distribution, multivariate Student distribution,

**AMS 2000 Subject Classification**: 60F05, 60G50, 60G55, 62E20, 62G30

In many situations related to experimental data analysis, one often comes across the following phenomenon: although conventional reasoning based on the central limit theorem of probability theory concludes that the expected distribution of observations should be normal, instead, the statistical procedures expose the noticeable non-normality of real distributions. Moreover, as a rule, the observed non-normal distributions are more leptokurtic than the normal law, having sharper vertices and heavier tails. These situations are typical in the financial data analysis (see, e.g., Chapter 4 in [1] or Chapter 8 in [2] and references therein), in experimental physics (see, e.g., [3]), and other fields dealing with statistical analysis of experimental data. Many attempts were undertaken to explain this heavy-

tailedness. Most significant theoretical breakthrough is usually associated with the

asymptotic behavior of sample quantiles is also considered.
