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

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

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

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

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

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

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

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

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

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