subdistributions

Subdistributions are a concept in probability theory and statistics that generalize the notion of distributions. While a distribution describes the probability of a random variable taking on a specific value, a subdistribution provides a way to describe the probability of a random variable being less than or equal to a certain value. This concept is particularly useful in the study of stochastic processes and in the analysis of extreme values.

In the context of probability theory, a subdistribution function (SDF) is a function that assigns to each real number x the probability that a random variable is less than or equal to x. This is analogous to the cumulative distribution function (CDF), which assigns the probability that a random variable is less than or equal to x. However, subdistributions can also be defined for random variables that do not have a CDF, such as those with infinite mean or variance.

Subdistributions are closely related to the concept of tail sigma-algebras, which are sigma-algebras generated by the events that a random variable is less than or equal to a certain value. This connection allows for the study of subdistributions using tools from measure theory and functional analysis.

In recent years, subdistributions have gained attention in the field of financial mathematics, where they are used to model the distribution of asset returns. This is because subdistributions can capture the heavy tails and skewness often observed in financial data, which are not adequately described by traditional distributions.