Comments on “Generalized Box-Müller Method for Generating q-Gaussian Random Deviates”

The generalized Box-Müller algorithm provides a methodology for generating q-Gaussian random variates, which generalizes the Gaussian
$(q=1)$
. The parameter
$-\infty < {q}\le 3$
is related to the shape of the tail decay;
${q} < 1$
for compact-support including parabola
$\left ({{q}={\it{ 0}} }\right)$
;
$1 < {q}\le 3$
for heavy-tail including Cauchy
$\left ({{q}=2 }\right)$
. This addendum clarifies the transformation
${q}^{\prime }=\frac {3{q}-1}{{q}+1}$
within the algorithm is due to a difference in the dimensions d of the generalized logarithm and the generalized distribution. The transformation is clarified by the decomposition of
${q}=1+\frac {2\kappa }{1+{d}\kappa }$
, where the shape parameter
$-1 < \kappa \le \infty $
quantifies the magnitude
$\vphantom {^{R}}$
of the deformation from exponential. A simpler specification for the generalized Box-Müller algorithm is provided using the shape of the tail decay.