We consider a diffusion process X t smoothed with (small)sampling parameter ε. As in Berzin, León and Ortega(2001), we consider a kernel estimate $\widehat{\alpha}_{\varepsilon}$ with window h(ε) of afunction α of its variance. In order to exhibit globaltests of hypothesis, we derive here central limit theorems forthe Lp deviations such as \[ \frac1{\sqrt{h}}\left(\frac{h}\varepsilon\right)^{\frac{p}2}\left(\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p-\mbox{I E}\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p\right).\]

We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$ using a kernel estimate f n drawn from a sample X1,...,Xn . The estimate θn is defined as any x in {X1,...,Xn } such that $f_n(x)=\max_{i=1, \hdots,n} f_n(X_i)$ . It is shown that θn behaves asymptotically as any maximizer ${\hat{\theta}}_n$ of f n . More precisely, we prove that for any sequence $(r_n)_{n\geq 1}$ of positive real numbers such that $r_n\to\infty$ and $r_n^d\log n/n\to 0$ , one has $r_n\,\|\theta_n-{\hat{\theta}}_n\| \to 0$ in probability. The asymptotic normality of θn follows without further work.

We consider a generalization of the so-called divide andcolor model recently introduced by Häggström. We investigate thebehavior of the magnetization in large boxes of the lattice $\mathbb{Z}^d$ and its fluctuations. Thus, Laws of Large Numbers and CentralLimit Theorems are proved, both quenched and annealed. We showthat the properties of the underlying percolation process deeplyinfluence the behavior of the coloring model. In the subcriticalcase, the limit magnetization is deterministic and the CentralLimit Theorem admits a Gaussian limit. Conversely, the limitmagnetization is not deterministic in the supercritical case andthe limit of the Central Limit Theorem is not Gaussian, except inthe particular model with exactly two colors which are equallyprobable. We also prove a Central Limit Theorem for the size of the intersection of the infinite cluster with large boxes in supercritical bond percolation.