Assume that the origin of the system ${x}_{t+1}=f(x_t)$ is globally asympotially stable. Now, if we have two systems running ${x}_{t+1}=f(x_t)+w_t$ and $s_{t+1}=f(s_t)+w_t$ where $w_t$ is a bounded iid noise that is equally added to both systems for all time steps $t\geq 0$. However, the initial condition of these two systems are different; i.e., $s_{0}\neq x_0$. Can we claim that in steady state $x_t=s_t$? If not, can we find a tight bound on the difference $||x_t-s_t||$ for $t \to \infty$?
Any help will be appreciated.