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2014

Delyon, Bernard, Portier, François

Let $(X_1,\ldots,X_n)$ be an i.i.d.

sequence of random variables in $\mathbb{R}^d$, $d\geq 1$.
We show that, for any function $\varphi :\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, \[n^ {1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{\varphi(X_i)}{\widehat{f}^(X_i)}- \int \varphi(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,\] where $\widehat{f}$ is the classical kernel estimator of the density of $X_1$.
This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\widehat{f}=f$.

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