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
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
there is a cascading dependence in every level of the hierarchy, for all satisfies a Markov property with a transition kernel that can be written in terms of the 's,
where and
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