  In this paper we consider the relation between random sums and compositions
of different processes. In particular, for independent Poisson processes
$N_\alpha(t)$, $N_\beta(t)$, $t>0$, we show that $N_\alpha(N_\beta(t))
\overset{\text{d}}{=} \sum_{j=1}^{N_\beta(t)} X_j$, where the $X_j$s are
Poisson random variables. We present a series of similar cases, the most
general of which is the one in which the outer process is Poisson and the inner
one is a nonlinear fractional birth process. We highlight generalisations of
these results where the external process is infinitely divisible. A section of
the paper concerns compositions of the form $N_\alpha(\tau_k^\nu)$, $\nu \in
(0,1]$, where $\tau_k^\nu$ is the inverse of the fractional Poisson process,
and we show how these compositions can be represented as random sums.
Furthermore we study compositions of the form $\Theta(N(t))$, $t>0$, which can
be represented as random products. The last section is devoted to studying
continued fractions of Cauchy random variables with a Poisson number of levels.
We evaluate the exact distribution and derive the scale parameter in terms of
ratios of Fibonacci numbers.
