×

zbMATH — the first resource for mathematics

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)
Citations:
Zbl 0093.12301
PDF BibTeX XML Cite