In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, a telescoping Markov chain (TMC) is a vector-valued stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 that satisfies a Markov property

Markov property

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It was named after the Russian mathematician Andrey Markov....

 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

Markov chain

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...

with transition probability matrix

(2) there is a cascading dependence in every level of the hierarchy,

    for all

(3) satisfies a Markov property with a transition kernel that can be written in terms of the 's,


where and