Product Convolution of Generalized Subexponential Distributions

Assume that xi and eta are two independent random variables with distribution functions F-xi and F eta, respectively. The distribution of a random variable xi eta, denoted by F-xi circle times F-eta, is called the product-convolution of F-xi and F-eta. It is proved that F-xi circle times F-eta is a generalized subexponential distribution if F-xi belongs to the class of generalized subexponential distributions and eta is nonnegative and not degenerated at zero.