arXiv subjects
Publications and source records attributed to Emile Le Page.
The purpose of this article is to show that Theorems 2.2-2.5 from [1] apply to the product of random matrices considered by Grama, Le Page, and Peigné [2]. This allows us, in particular, to emphasize the general nature of the formulation of our theorems in [1] by showing that our assumptions are verified for previous models.
We consider a general multivariate affine stochastic recursion and the associated Markov chain on $\mathbb R^{d}$. We assume a natural geometric condition which implies existence of an unbounded stationary solution and we show that the large values of the associated stationary process follow extreme value properties of classical type, with a non trivial extremal index. We develop some explicit consequences such as convergence to Fr{é}chet's law or to an exponential law, as well as convergence to a stable law. The proof is based on a spectral gap property for the action of associated positive operators on spaces of regular functions with slow growth, and on the clustering properties of large values in the recursion.
Let $(Z_n)_{n\geqslant 0}$ be a branching process in a random environment defined by a Markov chain $(X_n)_{n\geqslant 0}$ with values in a finite state space $\mathbb X$ starting at $X_0=i \in\mathbb X$. We extend from the i.i.d. environment to the Markovian one the classical classification of the branching processes into critical and strongly, intermediate and weakly subcritical states. In all these cases, we study the asymptotic behaviour of the probability that $Z_n>0$ as $n\to+\infty$.
Let $(X_n)_{n\geq 0}$ be a Markov chain with values in a finite state space $\mathbb X$ starting at $X_0=x \in \mathbb X$ and let $f$ be a real function defined on $\mathbb X$. Set $S_n=\sum_{k=1}^{n} f(X_k)$, $n\geqslant 1$. For any $y \in \mathbb R$ denote by $τ_y$ the first time when $y+S_n$ becomes non-positive. We study the asymptotic behaviour of the probability $\mathbb P_x \left( y+S_{n} \in [z,z+a] \,,\, τ_y > n \right)$ as $n\to+\infty.$ We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order $n^{3/2}$ and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability $\mathbb P_x \left( τ_y = n \right)$ as $n\to+\infty$.
Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert)_{n\geq 1}$ behaves like a sum of i.i.d.\ r.v.'s and satisfies standard classical properties such as the law of large numbers, law of iterated logarithm and the central limit theorem. Denote by $\mathbb{B}$ the closed unit ball in $\mathbb{R}^{d}$ and by $\mathbb{B}^{c}$ its complement. For any $v\in \mathbb{B}^{c}$ define the exit time of the random process $G_{n}v$ from $\mathbb{B}^{c}$ by $τ_{v}=\min \{n\geq 1:G_{n}v\in \mathbb{B}\} .$ We establish the asymptotic as $n \to \infty $ of the probability of the event $\{τ_{v}>n\} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}} \log \Vert G_{n}v\Vert $ conditioned that $τ_{v}>n.$
Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $μ$ (resp $λ$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp $H= \Aff (V))$ and assume that $μ$ is the projection of $λ$ on $G$. We study asymptotic properties of the iterated convolutions $μ^n *δ\_{v}$ (resp $λ^n*δ\_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $μ$ (resp $λ$), if the subsemigroup $T\subset G$ (resp.\ $Σ\subset H$) generated by the support of $μ$ (resp $λ$) is "large". We show spectral gap properties for the convolution operator defined by $μ$ on spaces of homogeneous functions of degree $s\geq 0$ on $V$, which satisfy H{ö}lder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $Σ\_{0}^{\infty} μ^k * δ\_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $λ$-boundary; then we use the above results and radial Fourier Analysis on $V\setminus \{0\}$ to show that the unique $λ$-stationary measure $ρ$ 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 $μ$ and $Σ$. 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 $λ$-boundary dual to $V$.
We consider a general multidimensional affine recursion with corresponding Markov operator $P$ and a unique $P$-stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of $P$. Spectral gap properties of $P$ and homogeneity at infinity of the $P$-stationary measure play an important role in the proofs.