## Approximating class approach for empirical processes of dependent sequences indexed by functions.(English)Zbl 1307.60027

The paper considers an empirical process of order $$n$$ given by ${U_n}(f) = {n^{ - {1 {\left/ {\vphantom{1 2}} \right. } 2}}}\sum_{i = 1}^n {(f({X_i}) - \int f\, d \mu )},$ where $$({X_i})$$ is a stationary process with values in a measurable space and with marginal distribution $$\mu$$, and $$f$$ belongs to a uniformly bounded class $$F$$ of real-valued measurable functions on this space. Given a Borel probability measure $$L$$ on $${l^\infty }(F)$$, $$({U_n}(f), \, n \geqslant 1)$$ is said to be convergent in distribution to $$L$$ if $${E^ * }(\varphi ({U_n})) \to \int {\varphi (x)\, dL(x)}$$ for all bounded and continuous real-valued functions on $${l^\infty }(F)$$, where $${E^ * }$$ denotes the outer integral. The main result of the paper gives conditions for the empirical process to converge in distribution in $${l^\infty }(F)$$ to a tight Gaussian process.

### MSC:

 60F17 Functional limit theorems; invariance principles 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60G10 Stationary stochastic processes 60G15 Gaussian processes 62E20 Asymptotic distribution theory in statistics
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### References:

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