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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References:

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