Why intervals? A simple limit theorem that is similar to limit theorems from statistics

Abstract

What is the set of possible values of a measurement error? In the majority of practical applications, an error is caused not by a single cause; it is caused by a large number of independent causes, each of which adds a small component to the total error. This fact is widely used in statistics: Namely, since it is known that the distibution of the sum of many independent small random variables is close to one of the so-called infinitely distribution of the total error is infinitely divisible. This assumption is used in the majority of the statistical applications. In this paper, we prove a similar result for the set of possible values of an error. Namely, if an error equals the sum of many small independent components, then its set of possible values is close to an interval; the smaller the components, the closer this set to an interval. This result provides one more justification for using intervals in data processing.