Summary
We consider a Markov chain on (E, ℬ) generated by a Markov kernel P. We study the question, when we can find for two initial distributions ν and μ two randomized stopping times T of (νX n ) n∈N and S of ( μ X n ) n∈N , such that the distribution of ν X T equals the one of μ X S and T, S are both finite.
The answer is given in terms of 〈ν-μ, h〉 with h bounded harmonic, or in terms of \(\mathop {\lim }\limits_{n \to \infty } \left\| {\frac{1}{{n + 1}} \cdot \sum\limits_0^n {(v - \mu )P^k } } \right\|\).
For stopping times T, S for two chains ( ν X n ) n∈N ,( μ X n ) n∈N we consider measures η, ζ on (E, ℬ) defined as follows: η(A)=expected number of visits of ( ν X n ) toA before T, ζ(A)=expected number of visits of (μ X n ) toA before S.
We show that we can construct T, S such that η and ζ are mutually singular and ℒ( v X T )=ℒ(μ X S . We relate η and ζ to the positive and negative part of certain solutions of the Poisson equation (I-P)(·)=ν-μ.
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Greven, A. Couplings of markov chains by randomized stopping times. Probab. Th. Rel. Fields 75, 195–212 (1987). https://doi.org/10.1007/BF00354033
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DOI: https://doi.org/10.1007/BF00354033
Keywords
- Markov Chain
- Stochastic Process
- Probability Theory
- Statistical Theory
- Poisson Equation