### A tail bound for sums of independent random variables and application to the Pareto distribution.

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We investigate the estimation of a multidimensional regression function $f$ from $n$ observations of an $\alpha $-mixing process $(Y,X)$, where $Y=f\left(X\right)+\xi $, $X$ represents the design and $\xi $ the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of $f$ in its construction) or it is supposed that $\xi $ is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework....

In this note we consider the estimation of the differential entropy of a probability density function. We propose a new adaptive estimator based on a plug-in approach and wavelet methods. Under the mean ${\mathbb{L}}_{p}$ error, $p\ge 1$, this estimator attains fast rates of convergence for a wide class of functions. We present simulation results in order to support our theoretical findings.

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