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The invariance principle for associated processes. (English) Zbl 0632.60001
Let \(\{X_ j\), \(j\in {\mathbb{N}}\}\) be a sequence of random variables, let \(S_ n=\sum^{n}_{1}X_ j\), \(\sigma^ 2_ n=E S^ 2_ n\), and let \(W_ n(t)=\sigma_ n^{-1}S_{[nt]}\) for \(t\in [0,1]\). Then \(\{X_ j\}\) is said to satisfy the invariance principle if \(W_ n\) converges weakly to standard Brownian motion on a specific set D of functions.
It is shown that associated sequences subject to certain moment conditions satisfy the invariance principle. Additional conditions ensuring asymptotic independence are required, although no stationarity is needed. Conditions on an associated sequence of random variables which imply the central limit theorem are also investigated.
Reviewer: A.Dale

60B10 Convergence of probability measures
60F05 Central limit and other weak theorems
Full Text: DOI
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