scholarly journals Limit theorems for semi-Markov processes

1966 ◽  
Vol 123 (2) ◽  
pp. 402-402 ◽  
Author(s):  
James Yackel
Keyword(s):  
Markov Processes ◽  
Limit Theorems ◽  
Semi Markov Processes

scholarly journals Limit theorems for semi-Markov processes

1978 ◽  
Vol 19 (2) ◽  
pp. 283-294 ◽  
Author(s):  
K.B. Athreya ◽  
P.E. Ney
Keyword(s):  
Markov Processes ◽  
Limit Theorems ◽  
General State ◽  
State Spaces ◽  
New Construction ◽  
Regeneration Times ◽  
Semi Markov Processes

A new construction of regeneration times is exploited to prove ergodic and renewal theorems for semi-Markov processes on general state spaces. This work extends results of the authors in Ann. Probability (6 (1978), 788–797).

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (03) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij (t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (3) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij(t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

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).

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