Limit theorems for J − X processes with a general state space

1976 ◽  
Vol 35 (1) ◽  
pp. 65-73 ◽  
Author(s):  
S. Grigorescu ◽  
G. OpriŞan
Keyword(s):  
State Space ◽  
Limit Theorems ◽  
General State ◽  
General State Space

Semi-Markov processes on a general state space: α-theory and quasi-stationarity

1980 ◽  
Vol 30 (2) ◽  
pp. 187-200 ◽  
Author(s):  
E. Arjas ◽  
E. Nummelin ◽  
R. L. Tweedie
Keyword(s):  
State Space ◽  
Markov Processes ◽  
Limit Theorems ◽  
General State ◽  
General State Space ◽  
Quasistationary Distributions ◽  
Positive Recurrent ◽  
Semi Markov Processes

AbstractBy amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

Criteria for classifying general Markov chains

1976 ◽  
Vol 8 (04) ◽  
pp. 737-771 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
The State ◽  
General State ◽  
Classical Work ◽  
The Status ◽  
General State Space

The aim of this paper is to present a comprehensive set of criteria for classifying as recurrent, transient, null or positive the sets visited by a general state space Markov chain. When the chain is irreducible in some sense, these then provide criteria for classifying the chain itself, provided the sets considered actually reflect the status of the chain as a whole. The first part of the paper is concerned with the connections between various definitions of recurrence, transience, nullity and positivity for sets and for irreducible chains; here we also elaborate the idea of status sets for irreducible chains. In the second part we give our criteria for classifying sets. When the state space is countable, our results for recurrence, transience and positivity reduce to the classical work of Foster (1953); for continuous-valued chains they extend results of Lamperti (1960), (1963); for general spaces the positivity and recurrence criteria strengthen those of Tweedie (1975b).

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