In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand : × $\to$ $\cup$ $\left\{\infty \right\}$, where is the -dimensional Euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

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.

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 $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.