Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A simple method to find all solutions to the functional equation of the
  smoothing transform   Given a nonincreasing null sequence $T = (T_j)_{j \ge 1}$ of nonnegative
random variables satisfying some classical integrability assumptions and
$\mathbb{E}(\sum_{j}T_{j}^{\alpha})=1$ for some $\alpha>0$, we characterize the
solutions of the well-known functional equation $$
f(t)\,=\,\textstyle\mathbb{E}\left(\prod_{j\ge 1}f(tT_{j})\right),\quad t\ge 0,
$$ related to the so-called smoothing transform and its min-type variant. In
order to do so within the class of nonnegative and nonincreasing functions, we
provide a new three-step method whose merits are that
  (1) it simplifies earlier approaches in some relevant aspects,
  (2) it works under weaker, close to optimal conditions in the so-called
boundary case when $\mathbb{E}\big(\sum_{j\ge 1}T_{j}^{\alpha}\log
T_{j}\big)=0$,
  (3) it can be expected to work as well in more general setups like random
environment.
  At the end of this article, we also give a one-to-one correspondence between
those solutions that are Laplace transforms and thus correspond to the fixed
points of the smoothing transform and certain fractal random measures. The
latter are defined on the boundary of a weighted tree related to an associated
branching random walk.

20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom