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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telescoping, markov, chain, this, article, does, cite, sources, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, september, 2010, learn. This article does not cite any sources Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Telescoping Markov chain news newspapers books scholar JSTOR September 2010 Learn how and when to remove this template message 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 gt 1 displaystyle N gt 1 consider the set of spaces S ℓ ℓ 1 N displaystyle mathcal S ell ell 1 N The hierarchical process 8 k displaystyle theta k defined in the product space 8 k 8 k 1 8 k N S 1 S N displaystyle theta k theta k 1 ldots theta k N in mathcal S 1 times cdots times mathcal S N is said to be a TMC if there is a set of transition probability kernels L n n 1 N displaystyle Lambda n n 1 N such that 8 k 1 displaystyle theta k 1 is a Markov chain with transition probability matrix L 1 displaystyle Lambda 1 P 8 k 1 s 8 k 1 1 r L 1 s r displaystyle mathbb P theta k 1 s mid theta k 1 1 r Lambda 1 s mid r there is a cascading dependence in every level of the hierarchy P 8 k n s 8 k 1 n r 8 k n 1 t L n s r t displaystyle mathbb P theta k n s mid theta k 1 n r theta k n 1 t Lambda n s mid r t for all n 2 displaystyle n geq 2 8 k displaystyle theta k satisfies a Markov property with a transition kernel that can be written in terms of the L displaystyle Lambda s P 8 k 1 s 8 k r L 1 s 1 r 1 ℓ 2 N L ℓ s ℓ r ℓ s ℓ 1 displaystyle mathbb P theta k 1 vec s mid theta k vec r Lambda 1 s 1 mid r 1 prod ell 2 N Lambda ell s ell mid r ell s ell 1 where s s 1 s N S 1 S N displaystyle vec s s 1 ldots s N in mathcal S 1 times cdots times mathcal S N and r r 1 r N S 1 S N displaystyle vec r r 1 ldots r N in mathcal S 1 times cdots times mathcal S N dd This probability related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Telescoping Markov chain amp oldid 877965802, wikipedia, wiki, book, books, library,