value for large $x$, therefore, the tail is a [*power-law*]{}. The asymptotic behavior of $\Theta(x)$ for small $x$, depends on $s$.

[**B.1**]{}: For $s>1/2$, it behaves as an [*exponential*]{} of $-1/x$, that dominates the low $x$ behavior of the PDF.

[**B.2**]{}: For $-1/2<s\le 0$, $\Theta(x)$ behaves as an [*exponential*]{} of $-1/x^{1+2s}$, that dominates the asymptotic behavior.

[**B.3**]{}: However, $\Theta(x)$ takes asymptotically a finite value, if $s<-1/2$; hence, the complete expression increases at the origin as a [*power-law*]{}.

At the threshold value $s=-1/2$, by employing again the explicit expression for $_2F_1(1,1,2,-z)$, one obtains $$\label{B}
\rho_{\rm B}(x) = \rho_o\frac{x^{\gamma_\mu\theta+1}}
{[1+x/\lambda^2]^{\gamma_\alpha+\gamma_\mu\theta+1} },  
\;\;\;\;\mbox{if $\gamma_\alpha>1$}\; .$$ Thus, the case $r=-1$, $s\le -1/2$ also provides twofold power-law distributions.

![ Diagram of the asymptotic behavior of the PDF defined by Eq. (\[FPss\]), in $(s,r)$-plane. Unshadowed regions and dotted borders are regions excluded by the normalization requirement. At both positive semi-axes, the growth at the origin is power law. On dark gray lines, tails are power law, with the tilted line corresponding to $r=2s-1$ ($p=-1$). Dashed lines correspond to pure exponential tails, with the tilted line corresponding to $r=2s$ ($p=0$). In the formulas, $a>0$, as well as the power-law exponents, generically depend on model parameters, moreover $p\equiv 