Étant donné un semi-flot mesurable $({\theta }_x{)}_{x\in {ℝ}_{+}^d}$ préservant une mesure de probabilité $\mu$ sur un espace $\Omega$, nous considérons les moyennes ergodiques $t^{-d}{\int }_{{ℝ}_{+}^d}\varphi (x/t)f\circ {\theta }_{x}dx$ où $\varphi$ est un “poids” à support compact sur ${ℝ}_{+}^d$, c’est-à-dire que $\varphi$ vérifie $\varphi \ge 0$ et $\int \varphi \left(x\right)dx=1$. Nous démontrons la convergence p.p. de ces moyennes quand $t\to +\infty$ si $f$ appartient à l’espace de Lorentz défini par le poids ${\varphi }^{*}$ qui est le réarrangé décroissant de $\varphi$. En particulier, pour $d=1$, on obtient la convergence p.p. des moyennes de Césarò d’ordre $\alpha$

The concept of robustness given by Zieliński (1977) is considered in cases where violations of models are generated by weight functions. Uniformly most bias-robust estimates of the scale parameter, based on order statistics, are obtained for some statistical models. Extensions of results of Zieliński (1983) and Bartoszewicz (1986) are given.

We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result has various...