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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
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