Citation
Hemstead, Robert Jack (1968) Stationary Absolute Distributions for Chains of Infinite Order. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/FXKFR517. https://resolver.caltech.edu/CaltechTHESIS:12072015112138031
Abstract
Let {Ƶ_{n}}^{∞}_{n = ∞} be a stochastic process with state space S_{1} = {0, 1, …, D – 1}. Such a process is called a chain of infinite order. The transitions of the chain are described by the functions
Q_{i}(i^{(0)}) = Ƥ(Ƶ_{n} = i  Ƶ_{n  1} = i ^{(0)}_{1}, Ƶ_{n  2} = i ^{(0)}_{2}, …) (i ɛ S_{1}), where i^{(0)} = (i^{(0)}_{1}, i^{(0)}_{2}, …) ranges over infinite sequences from S_{1}. 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 Q_{i}(i^{(0)}) such that
(i) 0 ≤ Q_{i}(i^{(0}) ≤ ξ ˂ 1
(ii)D – 1/Ʃ/i = 0 Q_{i}(i^{(0)}) Ξ 1
(iii) Q_{i}(i^{(n)}) → Q_{i}(i^{(0)}) whenever i^{(n)} → i^{(0)},
we prove the existence of a stationary chain of infinite order {Ƶ_{n}} whose transitions are given by
Ƥ (Ƶ_{n} = i  Ƶ_{n  1}, Ƶ_{n  2}, …) = Q_{i}(Ƶ_{n  1}, Ƶ_{n  2}, …)
With probability 1. The method also yields stationary chains {Ƶ_{n}} 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), 707724].
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)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  1 April 1968  
Funders: 
 
Record Number:  CaltechTHESIS:12072015112138031  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:12072015112138031  
DOI:  10.7907/FXKFR517  
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  
Last Modified:  02 Apr 2024 18:05 
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