# Functions of a Markov Process that are Markovian

• Richard A. Davis
• Keh-Shin Lii
• Dimitris N. Politis
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

## Summary

In this paper we are primarily concerned with discrete time parameter Markov processes {X(n)}, n = 0, 1, 2, …, with stationary transition mechanism. The processes {Y(n)} = {f(X(n))} generated by a given many-one function f and the processes {X(n)} with a fixed stationary transition mechanism are constructed. The processes {Y(n)} are in a one-to-one correspondence with the possible initial distributions of {X(n)}. The object of the paper is to determine conditions under which {Y(n)} is Markovian, whatever the initial distribution of {X(n)}. Necessary and sufficient conditions for the new processes {Y(n)} to be Markovian are obtained under the assumption that the family of measures corresponding to the fixed transition mechanism (of {X(n)}) is dominated [4]. The conditions are expressed, of course, in terms of the function f(•) and the transition mechanism. Generally the processes {Y(n)} do not have stationary transition mechanism. The conditions simplify in the case of a continuous time parameter Markov chain. Some of the discussions may at times have the flavor of those used in considering the concept of sufficiency [4]. Some aspects of the problem discussed in the paper are touched on in [2].

## Keywords

Probability Measure Markov Process Transition Function Initial Distribution Transition Mechanism
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## References

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## Authors and Affiliations

• Richard A. Davis
• 1
• Keh-Shin Lii
• 2
• Dimitris N. Politis
• 3
1. 1.Department of StatisticsColumbia UniversityNew YorkUSA
2. 2.Department of StatisticsUniversity of CaliforniaRiversideUSA
3. 3.Department of MathematicsUniversity of California, San DiegoLa JollaUSA