## 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. Giné 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

### Citations:

Zbl 0509.60012; Zbl 0553.60037; Zbl 0496.60004
Full Text: