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RELATIONS BETWEEN ERGODICITY AND MEAN DRIFT FOR MARKOV CHAINS 1
Author(s) -
Tweedie Richard L.
Publication year - 1975
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1975.tb00945.x
Subject(s) - infimum and supremum , aperiodic graph , markov chain , ergodicity , mathematics , stationary distribution , stationary state , bounded function , combinatorics , limit (mathematics) , distribution (mathematics) , state (computer science) , position (finance) , markov process , discrete mathematics , mathematical analysis , physics , statistics , quantum mechanics , algorithm , finance , economics
If {X n } is an irreducible aperiodic Markov chain on a state apace denote the mean one step change of position, or “drift”, of {X n } at j. The main result of this note is to show that, when |µ(j)| is bounded, {X n } admits a stationary distribution {π j }if and only if for some N > 0 and some state i, lim inf ∑when this holds, the limit infimum is in fact . Many of the known sufficient or necessary criteria for the existence of a stationary distribution can then be derived easily from this and related results.