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.[1]

For any N>1 consider the set of spaces {𝒮ℓ}ℓ=1N. The hierarchical process θk defined in the product-space

θk=(θk1,…,θkN)∈𝒮1×⋯×𝒮N

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

1. θk1 is a Markov chain with transition probability matrix Λ1

ℙ(θk1=s∣θk−11=r)=Λ1(s∣r)
2. there is a cascading dependence in every level of the hierarchy,

ℙ(θkn=s∣θk−1n=r,θkn−1=t)=Λn(s∣r,t) for all n≥2.
3. θk satisfies a Markov property with a transition kernel that can be written in terms of the Λ's,

ℙ(θk+1=s→∣θk=r→)=Λ1(s1∣r1)∏ℓ=2NΛℓ(sℓ∣rℓ,sℓ−1)

where s→=(s1,…,sN)∈𝒮1×⋯×𝒮N and r→=(r1,…,rN)∈𝒮1×⋯×𝒮N.

## References

1. ↑ "Markov Chains". 2021-11-22. https://ak2316.user.srcf.net/files/ib-markov-chains/markov-chains.pdf.

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