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

$\theta_k = (\theta_k^1,.....,\theta_k^N)\in\mathcal S^1\times......\times\mathcal S^N$

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

(1) $\theta_k^1$ is a Markov chain with transition probability matrix $\Lambda^1$

$\mathbb P(\theta_k^1=s|\theta_{k-1}^1=r)=\Lambda^1(s|r)$

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

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

$\mathbb P(\theta_{k+1}=\vec s|\theta_k=\vec r)=\Lambda^1(s_1|r_1)\prod_{\ell=2}^N\Lambda^\ell(s_\ell|r_\ell,s_{\ell-1})$

where $\vec s = (s_1,\ldots,s_N)\in\mathcal S^1\times\cdots\times\mathcal S^N$ and $\vec r = (r_1,\ldots,r_N)\in\mathcal S^1\times\cdots\times\mathcal S^N.$