Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains

Gareth O. Roberts; Jeffrey S. Rosenthal

doi:10.1214/105051606000000510

VARIANCE BOUNDING MARKOV CHAINS, L2-UNIFORM MEAN ERGODICITY AND THE CLT

Stochastics and Dynamics ◽

10.1142/s0219493711003176 ◽

2011 ◽

Vol 11 (01) ◽

pp. 81-94 ◽

Cited By ~ 1

Author(s):

YVES DERRIENNIC ◽

MICHAEL LIN

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Markov Chains ◽

Additive Functional ◽

Finite Variance ◽

Geometric Ergodicity ◽

The Central Limit Theorem ◽

Harris Recurrence ◽

Mean Ergodicity ◽

Compact Abelian Groups

We prove that variance bounding Markov chains, as defined by Roberts and Rosenthal [31], are uniformly mean ergodic in L2 of the invariant probability. For such chains, without any additional mixing, reversibility, or Harris recurrence assumptions, the central limit theorem and the invariance principle hold for every centered additive functional with finite variance. We also show that L2-geometric ergodicity is equivalent to L2-uniform geometric ergodicity. We then specialize the results to random walks on compact Abelian groups, and construct a probability on the unit circle such that the random walk it generates is L2-uniformly geometrically ergodic, but is not Harris recurrent.

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Foster-type criteria for Markov chains on general spaces

Journal of Applied Probability ◽

10.1239/jap/1165505219 ◽

2006 ◽

Vol 43 (4) ◽

pp. 1194-1200 ◽

Cited By ~ 6

Author(s):

Brian H. Fralix

Keyword(s):

Markov Chain ◽

Markov Chains ◽

State Space ◽

Open Problem ◽

Stopping Times ◽

Geometric Ergodicity ◽

General State ◽

Harris Recurrence ◽

General State Space ◽

Drift Conditions

This paper establishes new Foster-type criteria for a Markov chain on a general state space to be Harris recurrent, positive Harris recurrent, or geometrically ergodic. The criteria are based on drift conditions involving stopping times rather than deterministic steps. Meyn and Tweedie (1994) developed similar criteria involving random-sized steps, independent of the Markov chain under study. They also posed an open problem of finding criteria involving stopping times. Our results essentially solve that problem. We also show that the assumption of ψ-irreducibility is not needed when stating our drift conditions for positive Harris recurrence or geometric ergodicity.

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Foster-type criteria for Markov chains on general spaces

Journal of Applied Probability ◽

10.1017/s0021900200002540 ◽

2006 ◽

Vol 43 (04) ◽

pp. 1194-1200 ◽

Cited By ~ 1

Author(s):

Brian H. Fralix

Keyword(s):

Markov Chain ◽

Markov Chains ◽

State Space ◽

Open Problem ◽

Stopping Times ◽

Geometric Ergodicity ◽

General State ◽

Harris Recurrence ◽

General State Space ◽

Drift Conditions

This paper establishes new Foster-type criteria for a Markov chain on a general state space to be Harris recurrent, positive Harris recurrent, or geometrically ergodic. The criteria are based on drift conditions involving stopping times rather than deterministic steps. Meyn and Tweedie (1994) developed similar criteria involving random-sized steps, independent of the Markov chain under study. They also posed an open problem of finding criteria involving stopping times. Our results essentially solve that problem. We also show that the assumption of ψ-irreducibility is not needed when stating our drift conditions for positive Harris recurrence or geometric ergodicity.

Download Full-text