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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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