Consider a random process described by the following linear dynamics:
$$ x_{k+1} = a x_k + n_k, $$ where $|a|<1$ and $n_k$s are i.i.d. standard normal distributed.
It is quite easy to prove that $x_k$ converges to a zero mean normal distribution with variance $1/(1-a^2)$.
However, if we consider the following process, $$ x_{k+1} = \begin{cases} a x_k + n_k, & if |x|<M\\\\ b x_k + n_k, & if |x|\geq M \end{cases}, $$ where $|b|<1$ is also stable. I think it is quite easy to show that $x_k$ still converges to some stationary distribution with a zero mean.
On the other hand, is there a way to characterize that the covariance of such a distribution, especially when $M$ is very large? For example, something like $$ \left|\lim_{k\rightarrow \infty} \mathbb Ex_k^2 - 1/(1-a^2)\right| \leq C_1\times \exp(-C_2M^2), $$ where $C_1$ and $C_2$ are some constants related to $a$ and $b$.
The reason for believing the above inequality is that when $M$ is very large, there is a very small probability for $x$ to exit the region $\\{|x|<M\\}$ (which I think should be related to the error function of normal distribution, although $x_k$ is not exactly normal distributed), and even if it exists the region, it will come back very quickly since $b$ is stable. However, I am having trouble to put it in a rigorous way.
