Martingales à valeurs fermées bornées d’un espace métrique. (Martingales with values in closed bounded subsets of a metric space). (French) Zbl 0652.60015

Herer, Wojciech

C. R. Acad. Sci., Paris, Sér. I 305, 275-278 (1987).

We give [developing an idea of S. Doss,onal empirical processes for \(\Phi\)-mixing and a-mixing stochastic vectors to appropriate Gaussian processes under weaker conditions than K. J. Yoshihara [Z. Wahrscheinlichkeitstheor. Verw. Geb. 33, 133-137 (1975; Zbl 0304.60019)], i.e. either \[ \sum \Phi^{1/2}(2^ n)<\infty,\quad or\quad a(n)=O(n^{-r}), \] where \(r>2\), if \(p=1\); \(r>p+1\), if \(p\geq 2\) and even; \(r>p^ 2/(p-1)\), if \(p\geq 3\) and odd.

MSC:

60B99	Probability theory on algebraic and topological structures
60G44	Martingales with continuous parameter

Keywords:

theorems of convergence for martingales

Citations:

Zbl 0304.60019