Telescoping Markov chain

In probability theory, a telescoping Markov chain (TMC) is a vector-valued stochastic process that satisfies a Markov property and admits a hierarchical format through a network of transition matrices with cascading dependence.

For any $N>1$ consider the set of spaces $\{{\mathcal {S}}^{\ell }\}_{\ell =1}^{N}$. The hierarchical process $\theta _{k}$ defined in the product-space

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

is said to be a TMC if there is a set of transition probability kernels $\{\Lambda ^{n}\}_{n=1}^{N}$ such that

1. $\theta _{k}^{1}$ is a Markov chain with transition probability matrix $\Lambda ^{1}$

   $\mathbb {P} (\theta _{k}^{1}=s\mid \theta _{k-1}^{1}=r)=\Lambda ^{1}(s\mid r)$

2. there is a cascading dependence in every level of the hierarchy,

   $\mathbb {P} (\theta _{k}^{n}=s\mid \theta _{k-1}^{n}=r,\theta _{k}^{n-1}=t)=\Lambda ^{n}(s\mid r,t)$     for all $n\geq 2.$

3. $\theta _{k}$ satisfies a Markov property with a transition kernel that can be written in terms of the $\Lambda$'s,

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

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