Groupe d’études et de recherche en analyse des décisions

# Densities of sums and small ball probability

## Kwassi Joseph Dzahini

We propose a lemma that clarifies the proof of Theorem 4.1 on densities of sums in Rudelson and Vershynin. More precisely, by denoting by $$f_{S+Y}$$ the density of an absolutely continuous real-valued random variable $$S$$ augmented by an independent real-valued Gaussian random variable $$Y$$ with mean zero and an arbitrarily small variance, we prove that if $$f_{S+Y}$$ is bounded almost everywhere by a strictly positive constant $$C$$, then almost everywhere, the density $$f_S$$ is also bounded by the same constant $$C$$. Then, using these results, we show how small ball probability estimates such as $$\begin{equation*} ℙ{(|\sum_{k=1}^na_k\xi_k}|\leq\varepsilon)\leq C\varepsilon\quad\text{for all}\ \ \varepsilon>0, \end{equation*}$$ with $$a_k$$'s real numbers still hold when $$a_k$$'s are arbitrary random variables.

, 12 pages