= Explosive affine recursion with symmetric bounded noise
{title2=$X_{n+1}=aX_n+\theta_{n+1},\quad a>1,\quad\theta_n\in\{-1,1\}$}
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 $|X_{n+k}|\geq a^k(|X_n|-1/(a-1))+1/(a-1)$. 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.
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