A Generalization of Independence and Multivariate Student's t-distributions

In anomalous statistical physics, deformed algebraic structures are important objects. Heavily tailed probability distributions, such as Student's t-distributions, are characterized by deformed algebras. In addition, deformed algebras cause deformations of expectations and independences of random variables. Hence, a generalization of independence for multivariate Student's t-distribution is studied in this paper. Even if two random variables which follow to univariate Student's t-distributions are independent, the joint probability distribution of these two distributions is not a bivariate Student's t-distribution. It is shown that a bivariate Student's t-distribution is obtained from two univariate Student's t-distributions under q-deformed independence.