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

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 ) ∏ ℓ = 2 N Λ ℓ ( s ℓ | r ℓ , s ℓ − 1 )

where s → = ( s 1 , … , s N ) ∈ S 1 × ⋯ × S N and r → = ( r 1 , … , r N ) ∈ S 1 × ⋯ × S N .