arXiv 2003.093802020-03-20math.PR

关于$\mathbb R_+^d$上的仿射递归

On the affine recursion on $\mathbb R_+^d$

Sara Brofferio, Marc Peigné, Thi Da Cam Pham

摘要：

我们固定$d \geq 2$，并将$\mathcal S$表示为具有非负元素的$d \times d$矩阵的半群。我们考虑i的序列$(A_n, B_n)_{n \geq 1} $。I. D.随机变量的值在$\mathcal S\times \mathbb R_+^d$和研究的渐近行为的马尔可夫链$(X_n)_{n \geq 0}$对$ \mathbb R_+^d$定义为：\[ \forall n \geq 0, \qquad X_{n+1}=A_{n+1}X_n+B_{n+1}, \]其中$X_0$是一个固定的随机变量。我们假设矩阵$A_n$的李雅普诺夫指数等于$0$，并证明，在相当一般的假设下，存在一个唯一的（无限的）Radon措施$λ$ $(\mathbb R^+)^d$这是不变的链$(X_n)_{n \geq 0}$。$λ$的存在依赖于T.D.C. Pham最近关于随机矩阵乘积范数的波动的工作。它的唯一性是一个一般性质的结果，称为“局部收缩性”，强调了约20年前由M。Babillot，Ph. Bougerol et L.在一维仿射递归的情况下。

英文摘要：

We fix $d \geq 2$ and denote $\mathcal S$ the semi-group of $d \times d$ matrices with non negative entries. We consider a sequence $(A_n, B_n)_{n \geq 1} $ of i. i. d. random variables with values in $\mathcal S\times \mathbb R_+^d$ and study the asymptotic behavior of the Markov chain $(X_n)_{n \geq 0}$ on $ \mathbb R_+^d$ defined by: \[ \forall n \geq 0, \qquad X_{n+1}=A_{n+1}X_n+B_{n+1}, \] where $X_0$ is a fixed random variable. We assume that the Lyapunov exponent of the matrices $A_n$ equals $0$ and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure $λ$ on $(\mathbb R^+)^d$ which is invariant for the chain $(X_n)_{n \geq 0}$. The existence of $λ$ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices . Its unicity is a consequence of a general property, called "local contractivity", highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion .

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