Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral gap properties for linear random walks and Pareto's asymptotics
  for affine stochastic recursions   Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $\mu$ (resp
$\lambda$) a probability measure on the linear (resp affine) group $G=G L (V)$
(resp $H= \Aff (V))$ and assume that $\mu$ is the projection of $\lambda$ on
$G$. We study asymptotic properties of the iterated convolutions $\mu^n
*\delta\_{v}$ (resp $\lambda^n*\delta\_{v})$ if $v\in V$, i.e asymptotics of
the random walk on $V$ defined by $\mu$ (resp $\lambda$), if the subsemigroup
$T\subset G$ (resp.\ $\Sigma \subset H$) generated by the support of $\mu$
(resp $\lambda$) is "large". We show spectral gap properties for the
convolution operator defined by $\mu$ on spaces of homogeneous functions of
degree $s\geq 0$ on $V$, which satisfy H{\"o}lder type conditions. As a
consequence of our analysis we get precise asymptotics for the potential kernel
  $\Sigma\_{0}^{\infty} \mu^k * \delta\_{v}$, which imply its asymptotic
homogeneity. Under natural conditions the $H$-space $V$ is a
$\lambda$-boundary; then we use the above results and radial Fourier Analysis
on $V\setminus \{0\}$ to show that the unique $\lambda$-stationary measure
$\rho$ on $V$ is "homogeneous at infinity" with respect to dilations
$v\rightarrow t v$ (for $t\textgreater{}0$), with a tail measure depending
essentially of $\mu$ and $\Sigma$. Our proofs are based on the simplicity of
the dominant Lyapunov exponent for certain products of Markov-dependent random
matrices, on the use of renewal theorems for "tame" Markov walks, and on the
dynamical properties of a conditional $\lambda$-boundary dual to $V$.

22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs