# Determine the transition kernel of a stationary time-homogeneous Markov chain

Let $$(X_n)_{n\in\mathbb N_0}$$ be a stationary time-homogeneous Markov chain with $$X_0\sim f\mu$$ for some measure space $$(E,\mathcal E,\mu)$$ and $$\mathcal E$$-measurable $$f:E\to[0,\infty)$$ with $$\int f\:{\rm d}\mu=1.$$ We know that the evolution of $$(X_n)_{n\in\mathbb N_0}$$ is uniquely determined by its transition kernel, which is a Markov kernel $$\kappa$$ on $$(E,\mathcal E)$$ with $$\operatorname P\left[X_1\in B\mid X0\right]=\kappa(X_0,B)\;\;\;\text{almost surely for all }B\in\mathcal E\tag1.$$ Since we know that $$X_0,X_1\sim f\mu$$, are we able to determine $$\kappa$$ explicitly?

No, you cannot determine a kernel just from knowledge of the distributions of $$X_0$$ and $$X_1$$. You must know the joint distribution of $$(X_0,X_1)$$ to determine the kernel.
For example, consider $$X_0$$ and $$X_1$$ uniformly distributed on $$\{1,\ldots,n\}$$ and let $$\sigma$$ be any deterministic permutation. Then $$\sigma$$ induces a kernel sending $$i$$ to the dirac measure at $$\sigma(i)$$, and this kernel sends $$X_0$$ to $$X_1$$.
• If the chain is constructed by the Metropolis-Hastings algorithm (and we assume it already reached stationarity), then we should be able to determine $\kappa$, right? Aug 12 '19 at 6:44
• The acceptance criteria $a(x,y)$ is an arbitrary function in the notes you linked, and plays a role in determining the transition kernel. Note that "explicit" does not mean "uniquely determined". In any case, the discussion in these comments is tangential to the question you asked... Aug 12 '19 at 6:58