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Central limit theorem for Hölder processes on \(\mathbb R^m\)-unit cube. (English) Zbl 1199.60101
Let \(\beta \in (0,1)\). Consider the space \(C^{0,\beta }([0,1]^m)\) of all continuous functions \(f\) satisfying \(| f({\mathbf t})-f({\mathbf s})| \leq K\| {\mathbf t}-{\mathbf s}\| ^{\beta }\) for \({\mathbf s}, {\mathbf t} \in [0,1]^m\) where \(K\) is a constant. The main result of the paper can be formulated as follows. Let \((X_n({\mathbf t}), {\mathbf t}\in [0,1]^m)\) be a sequence of stochastic processes with continuous trajectories. Assume that the finite dimensional distributions of \((X_n)\) converge weakly and that \(\operatorname {Pr}(| X_n({\mathbf t})-X_n({\mathbf s})| \geq \varepsilon ) \leq K\varepsilon ^{-\alpha }\| {\mathbf s}-{\mathbf t}\| ^{\alpha \beta }\) for all \(\varepsilon >0\) and \({\mathbf s}, {\mathbf t} \in [0,1]^m\) where the constants \(\alpha \) and \(\beta \) satisfy some additional conditions. Then, the sequence \((X_n)\) converges weakly in the space \(C^{0,\gamma }([0,1]^m)\) where \(\gamma \) is a positive constant.

60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
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