This is, can we find an $m-$dependent ($1\leq m<\infty$) non-stationary, non-independent stochastic process $(X_{k})_{k}$ with the property that $\mathbb{P}[X_{k+1}\in A |X_{k}]=\nu(X_{k},A)$ for a fixed (independent on $k$) conditional probability distribution function $\nu:\mathbb{R}\times\mathcal{R}\to [0,1]$? ($\mathcal{R}$ is the Borel sigma algebra of $\mathbb{R}$).

If the answer is ''no'', can we do it if we require only that $E[X_{k+1}|X_{k}]=f\circ X_{k}$ for a fixed function $f:\mathbb{R}\to\mathbb{R}$ and every $k$?