# Telescoping Markov chain

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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 } = 1 N . The hierarchical process θ k defined in the product-space

θ k = ( θ k 1 , . . . . . , θ k N ) S 1 × . . . . . . × S N

is said to be a TMC if there is a set of transition probability kernels { Λ n } n = 1 N such that

1. θ k 1 is a Markov chain with transition probability matrix Λ 1 P ( θ k 1 = s | θ k 1 1 = r ) = Λ 1 ( s | r )
2. there is a cascading dependence in every level of the hierarchy,
3. θ k satisfies a Markov property with a transition kernel that can be written in terms of the Λ 's, P ( θ k + 1 = s | θ k = r ) = Λ 1 ( s 1 | r 1 ) = 2 N Λ ( s | r , s 1 )

where s = ( s 1 , , s N ) S 1 × × S N and r = ( r 1 , , r N ) S 1 × × S N .

## References

Telescoping Markov chain Wikipedia

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