×

Invariance principles for absolutely regular empirical processes. (English) Zbl 0817.60028

Let \((\xi_ i)_{i \in \mathbb{Z}}\) be a strictly stationary sequence of random elements of Polish space \(X\) with common distribution \(P\). Let \({\mathfrak F}_ 0 = \sigma (\xi_ i : i \leq 0)\) and \({\mathfrak G}_ n = \sigma (\xi_ i : i \geq n)\) and define the absolutely regular mixing coefficient \[ \beta_ n = {1 \over 2} \sup \sum_{(i,j) \in I \times J} \bigl | P(A_ i \cap B_ j) - P(A_ i) P(B_ j) \bigr | \] where the supremum is taken over all the finite partitions \((A_ i)_{i \in I}\) and \((B_ j)_{j \in J}\) of \({\mathfrak F}_ 0\) and \({\mathfrak G}_ n\), respectively.
Suppose \((\xi_ i)_{i \in \mathbb{Z}}\) is the absolutely regular sequence with mixing coefficient \(\beta_ n\) satisfying the summability condition \(\sum^ \infty_{n=1} \beta_ n < \infty\). Define the mixing rate function \(\beta (t) = \beta_{[t]}\) \((t \geq 1)\) and \(\beta (t) = 1\) otherwise. For any numerical function \(f\) define a new norm for \(f\) by \[ \| f \|_{2, \beta} = \left[ \int^ 1_ 0 \beta^{-1} (u) \bigl[ Q_ f(u) \bigr]^ 2du \right]^{1/2} \] where \(\beta^{-1}\) denotes the càdlàg inverse of the monotonic function \(\beta (\cdot)\) and \(Q_ f\) denotes the quantile function of \(| f(\xi_ 0) |\). Let \({\mathcal F}\) be a class of numerical functions with \(\| f \|_{2, \beta} < \infty\). Let \(P_ n\) be the empirical probability measure \(P_ n = \sum^ n_{i=1} \delta_{\xi_ i}\) and define \(Z_ n = \sqrt n(P_ n - P)\). The authors prove that a functional invariance principle in the sense of Donsker holds for \(\{Z_ n(f) : f \in {\mathcal F}\}\) under some conditions. The results extend the previous result of the authors [ibid. 30, No. 1, 63-82 (1994; Zbl 0790.60037)].

MSC:

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

Citations:

Zbl 0790.60037
PDFBibTeX XMLCite
Full Text: Numdam EuDML