Doukhan, Paul; Massart, Pascal; Rio, Emmanuel The functional central limit theorem for strongly mixing processes. (English) Zbl 0790.60037 Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 1, 63-82 (1994). 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. Cited in 4 ReviewsCited in 74 Documents MSC: 60F17 Functional limit theorems; invariance principles 62G99 Nonparametric inference Keywords:Donsker-Prokhorov invariance principle; strictly stationary; strongly mixing sequence; strong mixing function; functional central limit theorem PDFBibTeX XMLCite \textit{P. Doukhan} et al., Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 1, 63--82 (1994; Zbl 0790.60037) Full Text: Numdam EuDML