Suppose $|a| < 1$, $Z_1,Z_2,\ldots$ i.i.d. with distribution $\mu$ and put $X_n=aX_{n-1}+\z_n$. Then $X_n$ is a Markov chain with respect to $\F_n=\s(Z_1,\ldots,Z_n)$. This chain is called an autoregressiv moving average process, ARMAP for short. Show that its Markov operator is given by
$$
Pf(y)=\E(ay+\z)=\int f(ay+z)\,\mu(dz)
$$
2. In case $\mu$ is standard normal we get:
$$
P^nf(y)=\int f\Big(a^ny+z\sqrt{\frac{1-a^{2n}}{1-a^2}}\Big)\,\mu(dz)~.
$$
By induction we have
$$
P^nf(y)
=\int\cdots\int f(a^ny+a^{n-1}z_1+\cdots+z_n)
\,\mu(dz_1)\ldots\,\mu(dz_n)
$$
If $\mu$ is standard normal, then $a^{n-1}Z_1+\cdots+Z_n$ has the same distribution as $(1+a^2+\cdots+a^{2(n-1)})^{1/2}Z_1$.