# stochastic processes – How to characterize the variance of a linear Gaussian system with switching?

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,
$$x_{k+1} = begin{cases} a x_k + n_k, & if |x|
where $$|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
$$left|lim_{krightarrow infty} mathbb Ex_k^2 – 1/(1-a^2)right| leq C_1times exp(-C_2M^2),$$
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 $$\{|x| (which I think should be related to the error function of normal distribution, although $$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.