Stable random variables: Convolution and reliability

In this work, first we obtain a representation of the convolution between two asymmetric stable random variables of different parameters in terms of the Fox HH-function. In the symmetric case we obtain, in terms of the Meijer GG-function, a computable expression of the convolution for rational stability indices. Physically, this density function can be interpreted as the probabilistic generalization of the Voigt profile function. Second, we obtain the reliability P(X>Y)P(X>Y) for symmetric stable random variables of different stability indices, scale and shift parameters. Graphical illustrations and error analysis are given for the convolution, and the reliability is tested by Monte Carlo simulation.