# zbMATH — the first resource for mathematics

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.

##### MSC:
 60F17 Functional limit theorems; invariance principles 60G50 Sums of independent random variables; random walks
##### Keywords:
Hölder space; tightness; weak convergence
Full Text: