Telescoping Markov chain

A telescoping Markov chain (TMC) is a vector-valued stochastic process that satisfies a Markov property both in a spatial sense and a temporal sense. For any $N>0$ consider the set of discrete spaces $\{{\mathcal {S}}^{\ell }\}_{\ell =1}^{N}$. The process $\theta _{k}$ defined in the product-space

$\theta _{k}=(\theta _{k}^{1},.....,\theta _{k}^{N})\in {\mathcal {S}}^{1}\times ......\times {\mathcal {S}}^{N}$

is said to be a TMC if there is a set of transition probability matrices $\{\Lambda ^{\ell }\}_{\ell =1}^{N}$ such that there is a spatial Markov property

$\mathbb {P} (\theta _{k}^{n}=s|\theta _{k-1}^{n}=r,\theta _{k}^{n-1}=t)=\Lambda ^{n}(s|r,t)$

$\mathbb {P} (\theta _{k+1}={\vec {s}}|\theta _{k}={\vec {r}})=\Lambda ^{1}(s_{1}|r_{1})\prod _{\ell =2}^{N}\Lambda ^{\ell }(s_{\ell }|r_{\ell },r_{\ell -1})$

where ${\vec {s}}=(s_{1},....,s_{N})\in {\mathcal {S}}^{1}\times ......\times {\mathcal {S}}^{N}$ and ${\vec {r}}=(r_{1},.....,r_{N})\in {\mathcal {S}}^{1}\times ......\times {\mathcal {S}}^{N}$.