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Rates of convergence for empirical processes of stationary mixing sequences. (English) Zbl 0802.60024

Take \(X_ i\) to be a stationary sequence of random variables; let \(P\) denote the measure of \(X_ 1\) and let \(P_ n\) denote the empirical measure of the first \(n\) \(X_ i\). This paper establishes rates of convergence (in probability) to 0 of \(M_ n= \sup\{ | P_ n f- Pf|\): \(f\in F\}\), where \(Pf\) \((P_ n f)\) is the expectation of \(f\) with respect to \(P\) \((P_ n)\), and \(F\) is a suitable class of functions. In the case of independent \(X_ i\) this has been studied extensively [e.g. E. GinĂ© and J. Zinn, Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037)], and a number of authors have addressed similar questions for weakly dependent sequences. This paper examines absolutely regular sequences (a condition between strong and \(\varphi\) mixing) which decay at a slow rate, \(\beta(k)= o(k^{-r})\) for \(0<r <1\). Under a random form of the metric entropy condition, it establishes that \(M_ n\) converges to 0 at a rate that depends on the rate at which the random metric entropy converges to 0 in probability and the original mixing rate. This is valid for bounded classes \(F\). For general classes satisfying an \(L_ 1\) “envelope” function condition, convergence to 0 is shown.
To prove the main results, the author establishes an approximation of \(P_ n\) by a similar empirical measure, \(Q_ n\), defined on a “block- wise” version of \(\{X_ i\}\). This version of \(\{X_ i\}\) consists of independent copies of \(\{X_ i\): \(i\leq \mu_ n\}\) placed end-to-end. Here \(\mu_ n\) is a sequence of integers tending to infinity. The established theory on independent variables can be applied to \(Q_ n\). Lemma 4.1 develops Lemma 3.2 of H. Dehling [Z. Wahrscheinlichkeitstheorie Verw. Geb. 63, 393-432 (1983; Zbl 0496.60004)], and shows that the difference \(| P_ n f- Q_ n f|\) can be controlled by the choice of \(\mu_ n\), the rate of mixing and a bound on \(| f|\). Lemma 4.2 controls the tails of the distribution of \(\sup \{P_ n f\): \(f\in F\}\). Lemma 4.3 relates the entropy condition on the original sequence to the sequence defining \(Q_ n\). The paper contains a 5-page appendix discussing a number of measurability issues connected to the proofs.

MSC:

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
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