Large Deviations for Empirical Estimators of the Stationary Distribution of a Semi-Markov Process with Finite State Space

2008 ◽  
Vol 37 (19) ◽  
pp. 3077-3089 ◽  
Author(s):  
Claudio Macci
Keyword(s):  
Markov Process ◽  
Large Deviations ◽  
State Space ◽  
Stationary Distribution ◽  
Finite State ◽  
Semi Markov Process ◽  
Empirical Estimators ◽  
Finite State Space

Quasi-stationary distributions of birth-and-death processes

1978 ◽  
Vol 10 (03) ◽  
pp. 570-586 ◽  
Author(s):  
James A. Cavender
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
Parameter Family ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Birth And Death Processes ◽  
Finite State ◽  
Finite State Space ◽  
Quasi Stationary Distribution

Letqn(t) be the conditioned probability of finding a birth-and-death process in statenat timet,given that absorption into state 0 has not occurred by then. A family {q1(t),q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

Quasi-stationary distributions of birth-and-death processes

1978 ◽  
Vol 10 (3) ◽  
pp. 570-586 ◽  
Author(s):  
James A. Cavender
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
Parameter Family ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Birth And Death Processes ◽  
Finite State ◽  
Finite State Space ◽  
Quasi Stationary Distribution

Let qn(t) be the conditioned probability of finding a birth-and-death process in state n at time t, given that absorption into state 0 has not occurred by then. A family {q1(t), q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

1991 ◽  
Vol 28 (04) ◽  
pp. 822-832 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Time Interval ◽  
Reliability Model ◽  
Finite State ◽  
The Laplace Transform ◽  
Semi Markov Process ◽  
Number Of Visits ◽  
Semi Markov Processes ◽  
Finite State Space

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB (t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB (t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB (t) as t →∞.

Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability

2001 ◽  
Vol 38 (1) ◽  
pp. 195-208 ◽  
Author(s):  
Sophie Bloch-Mercier
Keyword(s):  
Markov Process ◽  
State Space ◽  
Markov Processes ◽  
Hazard Rate ◽  
Repairable System ◽  
Reversed Hazard Rate ◽  
Up States ◽  
Finite State ◽  
Hazard Rate Ordering ◽  
Finite State Space

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

1991 ◽  
Vol 28 (4) ◽  
pp. 822-832 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Time Interval ◽  
Reliability Model ◽  
Finite State ◽  
The Laplace Transform ◽  
Semi Markov Process ◽  
Number Of Visits ◽  
Semi Markov Processes ◽  
Finite State Space

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t →∞.

Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability

2001 ◽  
Vol 38 (01) ◽  
pp. 195-208 ◽  
Author(s):  
Sophie Bloch-Mercier
Keyword(s):  
Markov Process ◽  
State Space ◽  
Markov Processes ◽  
Hazard Rate ◽  
Repairable System ◽  
Reversed Hazard Rate ◽  
Up States ◽  
Finite State ◽  
Hazard Rate Ordering ◽  
Finite State Space

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

Rate modulation in dams and ruin problems

1996 ◽  
Vol 33 (2) ◽  
pp. 523-535 ◽  
Author(s):  
Søren Asmussen ◽  
Offer Kella
Keyword(s):  
State Space ◽  
Release Rate ◽  
Risk Process ◽  
The State ◽  
Limiting Distribution ◽  
Process Conditions ◽  
Explicit Formulas ◽  
Rate Modulation ◽  
Finite State ◽  
Finite State Space

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

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