Methods for generating random variates with Polya characteristic functions

Polya has shown that real even continuous functions that are convex on (0,∞), for 1 t = 0, and decreasing to 0 as t → ∞ are characteristic functions. Dugué and Girault (1955) have shown that the corresponding random variables are distributed as $\text{Y}\text{Z}$ where Y is a random variable with density (2π)^?1$\text{(}\text{sin}\text{(}\text{x}\text{2}\text{)/(}\text{x}\text{2}\text{))}{}^2$, and Z is independent of Y and has distribution function 1 ? φ + tφ′, t > 0. This property allows us to develop fast algorithms for this class of distributions. This is illustrated for the symmetric stable distribution, Linnik's distribution and a few other distributions. We pay special attention to the generation of Y.