1.3 Simulation of $ \alpha $-stable Variables

The complexity of the problem of simulating sequences of $ \alpha $-stable random variables results from the fact that there are no analytic expressions for the inverse $ F^{-1}$ of the cumulative distribution function. The first breakthrough was made by Kanter (1975), who gave a direct method for simulating $ S_\alpha(1,1,0)$ random variables, for $ \alpha <1$. It turned out that this method could be easily adapted to the general case. Chambers, Mallows, and Stuck (1976) were the first to give the formulas.

The algorithm for constructing a standard stable random variable $ X\sim
S_\alpha(1,\beta,0)$, in representation (1.2), is the following (Weron; 1996):

Given the formulas for simulation of a standard $ \alpha $-stable random variable, we can easily simulate a stable random variable for all admissible values of the parameters $ \alpha $, $ \sigma$, $ \beta$ and $ \mu$ using the following property: if $ X\sim
S_\alpha(1,\beta,0)$ then

$\displaystyle Y=\begin{cases}
\sigma X+\mu, & \alpha \ne 1, \cr\cr
\sigma X+\frac{2}{\pi}\beta\sigma\ln\sigma + \mu, & \alpha=1,
\end{cases}$     (1.8)

is $ S_\alpha(\sigma,\beta,\mu)$. It is interesting to note that for $ \alpha = 2$ (and $ \beta = 0$) the Chambers-Mallows-Stuck method reduces to the well known Box-Muller algorithm for generating Gaussian random variables (Janicki and Weron; 1994). Although many other approaches have been proposed in the literature, this method is regarded as the fastest and the most accurate (Weron; 2004).