A convolution identity and more with illustrations

Abstract

We begin with a new, general, interesting, and useful identity ( Theorem 2.1) based on recursive convolutions. Next, we extend the basic message from Theorem 2.1 by generalizing it under broadened set of assumptions. Interesting examples are provided involving multivariate Cauchy as well as multivariate equi-correlated normal and equi-correlated t distributions. Then, we discuss yet another approach ( Theorem 4.1) that also works in the evaluation of the expectation of a ratio of suitable random variables. Example 4.3 especially stands out because it cannot be handled by the previous approaches, but Theorem 4.1 steps in to help.