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On the first-order Edgeworth expansion for a Markov chain. (English) Zbl 0770.60023

Let \(X_ 1\), \(X_ 2,\dots\) be a homogeneous Markov process with measurable state space \((X,{\mathcal B})\) and initial distribution \(\pi\), having a stationary distribution \(p\) so that the \(n\)-step transition probability \(p^{(n)}(x,A)\) converges exponentially fast to \(p(A)\), uniformly w.r. to \(x\in X\) and \(A\in{\mathcal B}\). Let \(Y_ i=f(X_ i)\) with \(E_ pY_ i=0\) and \(E_ p| Y_ i|^ 3<\infty\) and let \(\sigma^ 2\) be the asymptotic variance of \(Z_ n=n^{-1/2}(Y_ 1+\cdots+Y_ n)\). Then, under certain conditions we have the first-order Edgeworth expansion \[ P(Z_ n\leq\sigma x)=\Phi(x)+n^{- 1/2}\left\{{1\over 6}\mu_ 3\sigma^{-3}(1-x^ 2)-\sigma^{-1}\mu_ \pi\right\}\varphi(x)+o(n^{-1/2}), \] uniformly in \(x\), with \(\Phi\) and \(\varphi\) the standard normal d.f. and density, \(\mu_ \pi=\sum^ \infty_{k=1}E_ \pi Y_ k\) and \(\mu_ 3\) an ‘asymptotic third moment’ under \(p\). The conditions are finiteness of moments and conditional nonlatticity of \(Y_ 1\). The expansion was first proved by S. V. Nagaev [Theory Probab. Appl. 6(1961), 62-81 (1962); translation from Teor. Veroyatn Primen. 6, 67-86 (1961; Zbl 0116.106)]. The authors give a corrected proof and then show by examples that the conditions are not necessary. The theorem then is stated and proved under weaker conditions (for the moments: \(\int| f(u)|^ 3p(x,du)\leq M)\). As an application, the expansion is derived for the maximum likelihood estimator of a transition probability when \(X\) is discrete.
Reviewer: A.J.Stam (Winsum)

MSC:

60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces

Citations:

Zbl 0116.106
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