Explosive affine recursion with symmetric bounded noise ($X_{n+1}=a{X}_n+{\theta }_{n+1},a>1,{\theta }_n\in \{-1,1\}$)

For independent fair signs, $|X_n|\to \infty$ almost surely from every deterministic initial value. Outside $[-L,L]$ with $L>1/(a-1)$, the sign is preserved and $|X_{n+k}|\ge a^k(|X_n|-1/(a-1))+1/(a-1)$. To reach that region, choose $m$ so $(a^m-1)/(a-1)>L$. At the beginning of each $m$-step block, choose the sign of the current value; a block of that sign pushes the absolute value past $L$ and has conditional probability $2^{-m}$. The chance of remaining inside after $k$ blocks is at most $(1-2^{-m}{)}^k$. This proof avoids any assumption about a density of the limiting random series.