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 $N>0$ considers the set of discrete spaces $\{{\mathcal {S}}^{\ell }\}_{\ell =1}^{N}$. The process $\theta _{k}\in {\mathcal {S}}^{1}\times ......\times {\mathcal {S}}^{N}$ given by

$\theta _{k}=(\theta _{k}^{0},.....,\theta _{k}^{N})\in {\mathcal {S}}^{1}\times ......\times {\mathcal {S}}^{N}$

is said to be a TMC if is there is