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 $\{{{𝒮}}^{\ell }{\}}_{\ell =1}^N$. The hierarchical process ${\theta }_k$ defined in the product-space

${\theta }_k=({\theta }_k^1,\dots ,{\theta }_k^N)\in {{𝒮}}^1\times \cdots \times {{𝒮}}^N$

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

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

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

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

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

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

   $\mathbb{\mathbb{P}}({\theta }_{k+1}=\overrightarrow{s}\mid {\theta }_k=\overrightarrow{r})={\mathrm{\Lambda }}^1(s_1\mid r_1)\prod _{\ell =2}^N{\mathrm{\Lambda }}^{\ell }(s_{\ell }\mid r_{\ell },s_{\ell -1})$

where $\overrightarrow{s}=(s_1,\dots ,s_N)\in {{𝒮}}^1\times \cdots \times {{𝒮}}^N$ and $\overrightarrow{r}=(r_1,\dots ,r_N)\in {{𝒮}}^1\times \cdots \times {{𝒮}}^N.$