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Une démonstration simple des théorèmes de Kolmogorov, Donsker et Itô-Nisio. (A simple proof of Kolmogorov, Donsker and Itô-Nisio theorems). (French. Abridged English version) Zbl 0764.60008
Let $$H_ \alpha$$, $$\alpha\in(0,1]$$, be the space of continuous functions $$f$$ on $$[0,1]$$ such that $$f(0)=0$$ and $\sup_{\substack{s,t\in[0,1] \\ s\neq t}} | f(s) - f(t)|/| s - t|^ \alpha<\infty$ and let $$H_\alpha^0$$, $$\alpha\in(0,1)$$, be the space of those $$f\in H_\alpha$$, such that $$| f(s)-f(t)| = o(|s - t|^\alpha)$$ when $$|s - t|\to 0$$. Using Schauder basis and a theorem of Z. Ciesielski [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 8, 217–222 (1960; Zbl 0093.12301)], the authors prove an $$H_\alpha$$-version of the classical Kolmogorov theorem, an $$H_\alpha^0$$-version of Donsker’s theorem and an $$H_\alpha^0$$-version of the Itô-Nisio theorem.

MSC:
 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
Zbl 0093.12301