Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compositions, Random Sums and Continued Random Fractions of Poisson and
  Fractional Poisson Processes   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.

27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph