#Stable random variable: sampling and density function

The right parametrization of the stabledist R package in order to get the \(\alpha\)-stable distribution with Laplace transform

\[L(u) = e^{-u^\alpha},\quad u\geq 0,\]

is

*stable(n,alpha, beta = 1, gamma = gamma_fun(alpha), delta = 0, pm = 1)

where gamma_fun(alpha) function is defined by \(\big(\cos\frac{\pi\alpha}{2}\big)^{1/\alpha}\). To see this, use for instance Proposition 1.2.12 from Taqqu & Samorodnitsky, Stable Non-Gaussian Random Processes (1994).

Below are histogram obtained with \(\alpha\)-stable data generated by the stabledist package (in blue) and by Kanter’s method (see Devroye (2009)) in red. The density function from stabledist package is superimposed (black curve).

#Corresponding polynomially tilted unilateral stable random variable \(T_{\alpha,\beta}\).

##Histogram against true density

Histogram of polynomially tilted unilateral stable random variable \(T_{\alpha,\beta}\) data generated using Devroye (2009) method, against true density of an \(\alpha\)-stable \(g_\alpha\), from stabledist package.

##Check of relation in distribution

We now check the relation \[G_{1+\theta/\alpha}=^d \left(\frac{G_{1+\theta}}{T_{\alpha,\theta}}\right)^\alpha.\] The histogram in red is obtained by using the r.h.s., ie our random variable \(T_{\alpha,\theta}\), and it is compared to the density of the l.h.s.

The fits are OK.

#The random variable \(Z_{alpha,b}\) sampled according to Devroye (2009) is consistent with its density.