 Exponential invariant law of a random affine recursion

Let $U_n$ have the uniform distribution on $[0,1]$ and let $E_n$ have an exponential distribution of rate $\lambda >0$, with all these inputs independent. The random affine recursion $X_n=U_n(X_{n-1}+E_n)$ has an invariant exponential law of the same rate. Indeed, for independent exponential $X,E$, beta-gamma independence says that $V=X/(X+E)$ is uniform and independent of $W=X+E$. Thus an independent uniform $U$ gives $UW$ the same law as $VW=X$.

For any deterministic real initial value $c$, couple the recursion to a stationary copy. Synchronous contraction of a random affine recursion gives an expected difference $2^{-n}\mathbb{E}|c-X_0|\le 2^{-n}(|c|+{\lambda }^{-1})$. Hence the one-time laws converge to $\mathrm{Exp}(\lambda )$, independently of $c$. An invariant initialization keeps every one-time law unchanged while the chain continues to fluctuate.