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 $\_^N$. 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 $\_^N$ such that

# $\theta_k^1$ is a Markov chain with transition probability matrix $\Lambda^1$
#: $\mathbb P(\theta_k^1=s\mid\theta_^1=r)=\Lambda^1(s\mid r)$
# there is a cascading dependence in every level of the hierarchy,
#: $\mathbb P(\theta_k^n=s\mid\theta_^n=r,\theta_k^=t)=\Lambda^n(s\mid r,t)$     for all $n\geq 2.$
# $\theta_k$ satisfies a Markov property with a transition kernel that can be written in terms of the $\Lambda$'s,
#:$\mathbb P(\theta_=\vec s\mid \theta_k=\vec r) = \Lambda^1(s_1\mid r_1) \prod_^N \Lambda^\ell(s_\ell \mid r_\ell,s_)$

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

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Markov processes