Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parametrised branching processes: a functional version of Kesten \&
  Stigum theorem   Let $(Z_n,n\geq 0)$ be a supercritical Galton-Watson process whose offspring
distribution $\mu$ has mean $\lambda>1$ and is such that $\int x(\log(x))_+
d\mu(x)<+\infty$. According to the famous Kesten \& Stigum theorem,
$(Z_n/\lambda^n)$ converges almost surely, as $n\to+\infty$. The limiting
random variable has mean~1, and its distribution is characterised as the
solution of a fixed point equation. \par In this paper, we consider a family of
Galton-Watson processes $(Z_n(\lambda), n\geq 0)$ defined for~$\lambda$ ranging
in an interval $I\subset (1, \infty)$, and where we interpret $\lambda$ as the
time (when $n$ is the generation). The number of children of an individual at
time~$\lambda$ is given by $X(\lambda)$, where $(X(\lambda))_{\lambda\in I}$ is
a c\`adl\`ag integer-valued process which is assumed to be almost surely
non-decreasing and such that $\mathbb E(X(\lambda))=\lambda >1$ for all
$\lambda\in I$. This allows us to define $Z_n(\lambda)$ the number of elements
in the $n$th generation at time $\lambda$.
  Set $W_n(\lambda)= Z_n(\lambda)/\lambda^n$ for all $n\geq 0$ and $\lambda\in
I$. We prove that, under some moment conditions on the process~$X$, the
sequence of processes $(W_n(\lambda), \lambda\in I)_{n\geq 0}$ converges in
probability as~$n$ tends to infinity in the space of c\`adl\`ag processes
equipped with the Skorokhod topology to a process, which we characterise as the
solution of a fixed point equation.

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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph