### G-2017-68

# 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.

Published **August 2017**
,
12 pages