Probabilistic Independence and Joint Cumulants

Abstract

The assumption of probabilistic independence is commonly made in engineering analysis to gain mathematical tractability. Although the criteria of the independence of random variables have been presented in many different ways, such as in terms of probability‐distribution functions, mathematical expectations, etc., a criterion in terms of joint cumulants has not yet been properly described. This paper presents the relationship between probabilistic independence and the joint cumulants (i.e. semi‐invariants) of random variables. When a random variable is a linear or nonlinear combination of a number of independent random variables, the calculation for the statistics of this random sum is more convenient using cumulants than central moments or moments. Examples that show the convenience of applying cumulants are presented, including a simple proof for the central limit theorem, the calculation of statistics for the Morison wave force, and the statistics of wave profiles using second‐order stochastic Stake's wave theory. In addition, an example related to the response statistics of linear systems subjected to quadratic Gaussian excitations is presented.