Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Critical branching processes in random environment with immigration:
  survival of a single family   We consider a critical branching process in an i.i.d. random environment, in
which one immigrant arrives at each generation. We are interested in the event
$\mathcal{A}_i(n)$ that all individuals alive at time $n$ are offspring of the
immigrant which joined the population at time $i$. We study the asymptotic
probability of this event when $n$ is large and $i$ follows different
asymptotics which may be related to $n$ ($i$ fixed, close to $n$, or going to
infinity but far from $n$). In order to do so, we establish some conditional
limit theorems for random walks, which are of independent interest.

24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
19
The categories of this paper are: eess