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# Stationary Absolute Distributions for Chains of Infinite Order

## Citation

Hemstead, Robert Jack (1968) Stationary Absolute Distributions for Chains of Infinite Order. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/FXKF-R517. https://resolver.caltech.edu/CaltechTHESIS:12072015-112138031

## Abstract

Let {Ƶn}n = -∞ be a stochastic process with state space S1 = {0, 1, …, D – 1}. Such a process is called a chain of infinite order. The transitions of the chain are described by the functions

Qi(i(0)) = Ƥ(Ƶn = i | Ƶn - 1 = i (0)1, Ƶn - 2 = i (0)2, …) (i ɛ S1), where i(0) = (i(0)1, i(0)2, …) ranges over infinite sequences from S1. If i(n) = (i(n)1, i(n)2, …) for n = 1, 2,…, then i(n) → i(0) means that for each k, i(n)k = i(0)k for all n sufficiently large.

Given functions Qi(i(0)) such that

(i) 0 ≤ Qi(i(0) ≤ ξ ˂ 1

(ii)D – 1/Ʃ/i = 0 Qi(i(0)) Ξ 1

(iii) Qi(i(n)) → Qi(i(0)) whenever i(n) → i(0),

we prove the existence of a stationary chain of infinite order {Ƶn} whose transitions are given by

Ƥ (Ƶn = i | Ƶn - 1, Ƶn - 2, …) = Qin - 1, Ƶn - 2, …)

With probability 1. The method also yields stationary chains {Ƶn} for which (iii) does not hold but whose transition probabilities are, in a sense, “locally Markovian.” These and similar results extend a paper by T.E. Harris [Pac. J. Math., 5 (1955), 707-724].

Included is a new proof of the existence and uniqueness of a stationary absolute distribution for an Nth order Markov chain in which all transitions are possible. This proof allows us to achieve our main results without the use of limit theorem techniques.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics and Philosophy)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Minor Option:Philosophy
Thesis Availability:Public (worldwide access)
• Krieger, Henry A.
Thesis Committee:
• Unknown, Unknown
Defense Date:1 April 1968
Funders:
Funding AgencyGrant Number
Woodrow Wilson FoundationUNSPECIFIED
NSFUNSPECIFIED
CaltechUNSPECIFIED
Record Number:CaltechTHESIS:12072015-112138031
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:12072015-112138031
DOI:10.7907/FXKF-R517
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9309
Collection:CaltechTHESIS
Deposited By:INVALID USER
Deposited On:07 Dec 2015 21:25