Groupe d’études et de recherche en analyse des décisions

G-2017-68

Densities of sums and small ball probability

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