How to Describe Measurement Errors: A Natural Generalization of the Central Limit Theorem Beyond Normal (and Other Infinitely Divisible) Distributions

When precise measurement instruments are designed, designers try their best to decrease the effect of the main factors leading to measurement errors. As a result of this decrease, the remaining measurement error is the joint result of a large number of relatively small independent error components. According to the Central Limit Theorem, under reasonable conditions, when the number of components increases, the resulting distribution tends to Gaussian (normal). Thus, in practice, when the number of components is large, the distribution is close to normal -- and normal distributions are indeed ubiquitous in measurements. However, in some practical situations, the distribution is different from Gaussian. How can we describe such distributions? In general, the more parameters we use, the more accurately we can describe a distribution. The class of Gaussian distributions is 2-dimensional, in the sense that each distribution from this family can be uniquely determined by 2 parameters: e.g., mean and standard deviations. Thus, when the approximation of the measurement error by a normal distribution is insufficiently accurate, a natural idea is to consider families with more parameters. What are 3-, 4-, 5-, n-dimensional limit families of this type? Researchers have considered 3-dimensional classes of distributions, which can -- under weaker assumptions -- to describe similar limit cases; distributions from these families are known as infinitely divisible ones. A natural next question is to describe all possible n-dimensional families for all n. Such a description is provided in this paper