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A central limit theorem for functions of a Markov chain with applications to shifts. (English) Zbl 0762.60023

Let \(Q\) be a Markov kernel and \(\pi\) a probability measure on a Polish space \(S\) and suppose that \((X_ k)_{k\geq 0}\) is a strictly stationary and ergodic Markov chain with \(Q\) as transition function and \(\pi\) as initial distribution. Let \(\mathcal L\) denote the set of all \(\xi\in L^ 2(\pi)\) for which \(\int\xi d\pi=0\), and put for a given \(\xi\in{\mathcal L}\), \(S_ n(\xi)=\sum^ n_{k=1}\xi(X_ k)\). The main result of the paper is the following central limit theorem for \((X_ k)_{k\geq 0}\). Let \(\pi_ 1\) denote the joint distribution of \(X_ 0\) and \(X_ 1\) and suppose that the given \(\xi\in{\mathcal L}\) has the property that there exists a \(g\in L^ 2(\pi_ 1)\) for which \[ \lim_{n\to\infty}\sum^ n_{k=1}(Q^{k-1}\xi(y)-Q^ k\xi(x))=g(x,y)\tag{*} \] in \(L^ 2(\pi_ 1)\). Then \(n^{-1/2}(S_ n(\xi)-\mathbb{E}(S_ n(\xi)\mid X_ 0))\) is asymptotically normal. This result is applied to the Bernoulli and the Lebesgue shift. In both cases, conditions in terms of the Fourier coefficients of \(\xi\) are given which imply condition (*).

MSC:

60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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