that $X+Y_2$ has distribution $\mu(\alpha,\beta,\lambda)$, then by Proposition \[GIGInv\] we get that $(X+Y_2)^{-1}$ has distribution $\mu(\beta,\alpha,-\lambda)$. Then we have that $Y_1+(Y_2+X)^{-1}$ has the distribution $\mu(\beta,\alpha,\lambda)$ and finally we get $(Y_1+(Y_2+X)^{-1})^{-1}$ has the desired distribution $\mu(\alpha,\beta,-\lambda)$.

3.  Both identities above can be iterated finitely many times, so that one obtains that $X\stackrel{d}{=}\left(Y_1+\left(Y_2+\cdots\right)^{-1}\right)^{-1}$, where $Y_1,Y_2,\ldots$ are free, for $k$ odd $Y_k$ has the free Poisson distribution $\nu(1/\beta,\lambda)$ and for $k$ even $Y_k$ has the distribution $\nu(1/\alpha,\lambda)$. For the case described in $1^o$ one simply has to take $\alpha=\beta$. We are not sure if infinite continued fractions can be defined.

Next we study limits of the fGIG measure $\mu(\alpha,\beta,\lambda)$ when $\alpha\to 0$ and $\beta\to 0$. This was stated with some mistake in [@Szp17 Remark 2.3].

As $\beta\downarrow 0$ we have the following weak limits of the fGIG distribution $$\begin{aligned}
\label{eq:limits}
    \lim_{\beta\downarrow 0}\mu(\alpha,\beta,\lambda) = 
    \begin{cases} \nu(1/\alpha, \lambda), & \lambda \geq1, \\ 
    \frac{1-\lambda}{2}\delta_0 + \frac{1+\lambda}{2}\nu(\frac{1+\lambda}{2\alpha},1),  & |\lambda|<1,\\
    \delta_0, & \lambda\leq -1.  
    \end{cases}
    \end{aligned}$$ Taking into account Proposition \[GIGInv\] one can also describe limits when $\alpha\downarrow 0$ for $\lambda \geq1$.

This result reflects the fact that GIG matrix generalizes the Wishart matrix for $\lambda \geq1$, but not for $\lambda <1$ (see [@Fer06] for GIG matrix and 