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The functional central limit theorem for strongly mixing processes. (English) Zbl 0790.60037

Summary: Let \((X_ i)_{i\in\mathbb{Z}}\) be a strictly stationary and strongly mixing sequence of \(\mathbb{R}^ d\)-valued zero-mean random variables. Let \((\alpha_ n)_{n>0}\) be the sequence of mixing coefficients. We define the strong mixing function \(\alpha\) by \(\alpha(t)=\alpha_{[t]}\) and we denote by \(Q\) the quantile function of \(| X_ 0|\), which is the inverse function of \(t\to\mathbb{P}(| X_ 0|>t)\). The main result of this paper is that the functional central limit theorem holds whenever the following condition is fulfilled: \[ \int^ 1_ 0\alpha^{- 1}(t)[Q(t)]^ 2dt<\infty, \tag{*} \] where \(f^{-1}\) denotes the inverse of the monotonic function \(f\). Note that this condition is equivalent to the usual condition \(\mathbb{E}(X^ 2_ 0)<\infty\) for \(m\)-dependent sequences. Moreover, for any \(a>1\), we construct a sequence \((X_ i)_{i\in\mathbb{Z}}\) with strong mixing coefficients \(\alpha_ n\) of the order of \(n^{-a}\) such that the CLT does not hold as soon as condition \((^*)\) is violated.

MSC:

60F17 Functional limit theorems; invariance principles
62G99 Nonparametric inference
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