Selected Works of Murray Rosenblatt pp 134-145 | Cite as

# Functions of a Markov Process that are Markovian

## 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

Eter Radon Dition### References

- 1.R. R. Bahadur, Sufficiency and statistical decision functions,
*Ann. Math. Stat*, 25 (1954), pp. 423–462.MathSciNetCrossRefGoogle Scholar - 2.C. J. Burke & M. Rosenblatt, A Markovian function of a Markov chain,
*Ann. Math. Stat*., 29 (1958), pp. 1112–1122MathSciNetMATHCrossRefGoogle Scholar - 3.J. L. Doob,
*Stochastic Processes*, New York (1952).Google Scholar - 4.P. R. Halmos & L. J. Savage, Application of the Radon-Nikodym theorem to the theory of sufficient statistics,
*Ann. Math. Stat*, 20 (1949), pp. 225–241.MathSciNetMATHCrossRefGoogle Scholar - 5.M. Lqeve,
*Probability Theory*, New York (1955).Google Scholar - 6.B. Rankin,
*The concept of enchainment—a relation between stochastic processes*, unpublished (1955).Google Scholar - 7.D. Rosenblatt, On aggregation and consolidation in linear systems, to be published in the
*Naval Research Logistics Quarterly.*Google Scholar - 8.D. Rosenblatt,
*On aggregation and consolidation in linear systems*, Technical Report C, prepared under Contract Nonr-1180 (00) (NR-047–012) 1956.Google Scholar - 9.D. Rosenblatt, Abstracts of “On aggregation and consolidation in finite substochastic systems I, II, III, IV,”
*Ann. Math. Stat*, 28 (1957), pp. 1060–1061.Google Scholar - 10.M. Rosenblatt, Abstract of “Functions of Markov Chains,”
*Ann. Math. Stat.*, 29 (1958), p. 1291.MathSciNetGoogle Scholar