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 $k$?



Is there a Conditionally Stationary, non-stationary Process which is (strictly) $m-$Dependent?
Let $\{X\_i\}\_{i=0}^{\infty}$ be an i.i.d. Bernoulli sequence with $P[X\_i=1]=P[X\_i=0]=1/2$. Consider: 


$$\{X\_1, X\_0, X\_2, X\_1, X\_3, X\_2, X\_4, X\_3, X\_5, X\_4, X\_6, X\_5, ...\}$$ 


