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
consider the set of spaces
. The hierarchical process
defined in the product-space
![{\displaystyle \theta _{k}=(\theta _{k}^{1},\ldots ,\theta _{k}^{N})\in {\mathcal {S))^{1}\times \cdots \times {\mathcal {S))^{N))](https://wikimedia.org/api/rest_v1/media/math/render/svg/022a7e759e1a607803ab51ac2992855753c7db19)
is said to be a TMC if there is a set of transition probability kernels
such that
is a Markov chain with transition probability matrix
![{\displaystyle \mathbb {P} (\theta _{k}^{1}=s\mid \theta _{k-1}^{1}=r)=\Lambda ^{1}(s\mid r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf68af0cc6040bbc1f5fcccba8e651701a501a3b)
- there is a cascading dependence in every level of the hierarchy,
for all ![{\displaystyle n\geq 2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12579de3af09ac1e4dd0c0724536b2361760f498)
satisfies a Markov property with a transition kernel that can be written in terms of the
's,
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48a0ee8b46b022674f01063cabbff10a74c5f47c)
- where
and ![{\displaystyle {\vec {r))=(r_{1},\ldots ,r_{N})\in {\mathcal {S))^{1}\times \cdots \times {\mathcal {S))^{N}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84f2531b4cc0eee991d7e6930a920e47ebd33863)