Telescoping Markov chain


In probability theory, a telescoping Markov chain 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
  1. is a Markov chain with transition probability matrix
  2. :
  3. there is a cascading dependence in every level of the hierarchy,
  4. : for all
  5. satisfies a Markov property with a transition kernel that can be written in terms of the 's,
  6. :