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 [math]\displaystyle{ N\gt 1 }[/math] consider the set of spaces [math]\displaystyle{ \{\mathcal S^\ell\}_{\ell=1}^N }[/math]. The hierarchical process [math]\displaystyle{ \theta_k }[/math] defined in the product-space

[math]\displaystyle{ \theta_k = (\theta_k^1,\ldots,\theta_k^N)\in\mathcal S^1\times\cdots\times\mathcal S^N }[/math]

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

1. [math]\displaystyle{ \theta_k^1 }[/math] is a Markov chain with transition probability matrix [math]\displaystyle{ \Lambda^1 }[/math]

   [math]\displaystyle{ \mathbb P(\theta_k^1=s\mid\theta_{k-1}^1=r)=\Lambda^1(s\mid r) }[/math]

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

   [math]\displaystyle{ \mathbb P(\theta_k^n=s\mid\theta_{k-1}^n=r,\theta_k^{n-1}=t)=\Lambda^n(s\mid r,t) }[/math]     for all [math]\displaystyle{ n\geq 2. }[/math]

3. [math]\displaystyle{ \theta_k }[/math] satisfies a Markov property with a transition kernel that can be written in terms of the [math]\displaystyle{ \Lambda }[/math]'s,

   [math]\displaystyle{ \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}) }[/math]

where [math]\displaystyle{ \vec s = (s_1,\ldots,s_N)\in\mathcal S^1\times\cdots\times\mathcal S^N }[/math] and [math]\displaystyle{ \vec r = (r_1,\ldots,r_N)\in\mathcal S^1\times\cdots\times\mathcal S^N. }[/math]