New Issues About the Central Limit Theorem Based on a More Comprehensive Approach to Probability: Developments and Future Perspectives

A more comprehensive approach to probability studies an infinite number of probability
laws that are formally admissible. Thus, an infinite number of weighted averages can be handled.
Areinterpretation of the central limit theorem is accordingly shown. The deviations or errors from
a fixed value are calculated. It is proved that they are normally distributed. Furthermore, such
deviations are invariant with respect to geometric translations identifying repeated samples. In this
paper, the way of understanding the statistical model to which a specific and pragmatic distribution
is compared is not a functional scheme in the continuum, but it is itself a specific and pragmatic
distribution. It is possible to enlarge the reasoning, so developments and future perspectives that
underlie the reinterpretation of the central limit theorem are discussed.