r-Free Convolution and Variance Function

The concept of r-free convolution (which is represented by r) was introduced for 0≤r≤1. It is equal to the Boolean additive convolution ⊎ if r=0 and reduced to the free additive convolution ⊞ when r=1. This paper presents certain features of the r-free convolution in relation to the CSK families and their associated variance functions. We provide the variance function formula under r-convolution power. We then estimate members of the r-free Gaussian and r-free Poisson CSK families using the variance function and the r-convolution, respectively. Additionally, a novel limit theorem for the r-convolution is provided utilizing the variance functions and the free multiplicative convolution.