Let be a distribution function (d.f) in the domain of attraction of an extreme value distribution $H_{\gamma }$; it is well-known that , where is the d.f of the excesses over , converges, when tends to , the end-point of , to $G_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function which verifies ${lim}_{u\to s_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}(x/\sigma \left(u\right))|$ converges to faster than $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.

We consider an estimate of the mode of a multivariate probability density with support in ${ℝ}^d$ using a kernel estimate drawn from a sample . The estimate is defined as any in {} such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of . More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of follows without further work.

We investigate the minima of functionals of the form $${\int }_{[a,b]}g\left(\dot{u}\left(s\right)\right)\mathrm{d}s$$ where is strictly convex. The admissible functions $u:[a,b]⟶ℝ$ are not necessarily convex and satisfy $u\le f$ on , , , is a fixed function on . We show that the minimum is attained by $\overline{f}$, the convex envelope of .