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Convergence in total variation of an affine random recursion in $${[0, p)}^k$$ to a uniform random vector. (English) Zbl 1311.60012
The author studies the rate of convergence of the $$k$$-dimensional Markov chain $\mathbf{X}_{n+1}=A\mathbf{X}{}_{n}{}_+\mathbf{B}{}_{n}\mod p,$ where $$A$$ is an integer matrix, $$(\mathbf{B}_{n})$$ is a sequence of i.i.d. real random vectors, and $$p>0$$. The variation distance of two probability measures $$\varphi$$ and $$\psi$$ on the measurable space $$(E,\mathcal{E})$$ is defined by $\|\varphi-\psi\|:=\sup_{A\in\mathcal{E}}|\varphi(A)-\psi(A)|.$ Put $\operatorname{P}_{x_0}^{n}(A)=\operatorname{P}(X_n\in A\mid X_0=x_0);$ then $$(X_n)$$ is
(a)
$$\varphi$$-irreducible if there exists a measure $$\varphi$$ on $$(E,\mathcal{E})$$ such that, for all $$A\in\mathcal{E}$$ with $$\varphi(A)>0$$ and $$x_0\in E$$, there exists $$n=n(x_0,A)$$ such that $$\operatorname{P}_{x_0}^{n}(A)>0$$,
(b)
uniformly ergodic if $$\lim_{n\to+\infty} \sup_{x_0\in E} \|\operatorname{P}_{x_0}^{n}-\pi\|=0$$, where $$\pi$$ is a probability measure on $$(E,\mathcal{E})$$.

The author proves that $$(X_n)$$ is uniformly ergodic for $$\pi=\mathcal{L}_{\mathbf{U}}$$, the uniform distribution on $$[0,p)^k$$.

##### MSC:
 60B10 Convergence of probability measures 60J05 Discrete-time Markov processes on general state spaces