Let $\{\xi_i\}_{i\geq 0}$ be a sequence of iid random variables that are uniform on a d-dimensional box $B_1(0) = [-1,1]^d$. Let $\{A_i\}:\mathbb{R}^d \to \mathbb{R}^d$ be invertible matrices with such that $\|A_i\|<\lambda^i$ for $\lambda<1$ (one can think that $A_i$ is just the identity matrix $Id$ times $\lambda^i$).
Consider random variable $X = \sum_{i=0}^{\infty} A_i \xi_i$. It is clear that there exists a set $V$ such that $B_1(0)\subset V$ and the distribution of $X$ has a positive density on $V$ (if $A_i = \lambda \cdot Id$, then $V$ is the $1/(1-\lambda)$-neighborhood of $B_1(0)$).
Let $W \in B_\epsilon(0)$ be another random variable on the same probability space, where $\epsilon$ is small, say $\epsilon<1/1000$. Now consider random variable $Z=X+W$.
In other words, I have to functions $f_X,f_W:([-1,1]^d)^\mathbb{N} \to \mathbb{R}^d$, I know how $f_X$ looks like and I know that $Image(f_W)\subset B_\epsilon (0)$. I consider $f_Z=f_X+f_W$. I know that $f_W$ does depend in a nontrivial way on all the coordinates i.e. $X$ and $W$ are not independent.
I have two questions
- What formal argument can I use to prove that there exists an open set $U$ such that the distribution of $Z$ has a positive density on $U$? I expect that there is some very classical lemma to do it.
- Is it possible to write an estimate for the measure of $U \cap V$ (in terms of $\epsilon$)?