Limit Theorems for Stopped Functionals of Markov Renewal Processes

1999 ◽  
Vol 51 (2) ◽  
pp. 369-382 ◽  
Author(s):  
Gerold Alsmeyer ◽  
Allan Gut
Keyword(s):  
Limit Theorems ◽  
Renewal Processes ◽  
Markov Renewal Processes

scholarly journals Limit Theorems for Markov Renewal Processes

1964 ◽  
Vol 35 (4) ◽  
pp. 1746-1764 ◽  
Author(s):  
Ronald Pyke ◽  
Ronald Schaufele
Keyword(s):  
Limit Theorems ◽  
Renewal Processes ◽  
Markov Renewal Processes

On non-singular Markov renewal processes with an application to a growth–catastrophe model

1985 ◽  
Vol 22 (02) ◽  
pp. 253-266
Author(s):  
Seppo Niemi
Keyword(s):  
Markov Process ◽  
Total Variation ◽  
Population Model ◽  
Limit Theorems ◽  
Renewal Processes ◽  
Regenerative Processes ◽  
Catastrophe Model ◽  
Markov Renewal Processes ◽  
Harris Recurrence ◽  
Forward Process

The paper is concerned with Markov renewal processes satisfying a certain non-singularity condition. The relation of this condition to irreducibility, Harris recurrence and regularity of the associated forward Markov process is studied. This enables one to prove limit theorems of a total variation type for Markov renewal processes and semi-regenerative processes by applying Orey's theorem to the forward process. The results are applied to a GI/G/1 queue and a growth-catastrophe population model.

Central limit theorems for multivariate semi-Markov sequences and processes, with applications

1999 ◽  
Vol 36 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Central Limit ◽  
Limit Theorems ◽  
Markov Models ◽  
Central Limit Theorems ◽  
Renewal Processes ◽  
Counting Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Processes

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

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.

scholarly journals Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 158
Author(s):  
Anatoliy Swishchuk ◽  
Nikolaos Limnios
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Diffusion Approximation ◽  
Limit Theorems ◽  
Stochastic Systems ◽  
Renewal Processes ◽  
Additive Functionals ◽  
Markov Random ◽  
Markov Renewal Processes ◽  
Random Evolutions

In this paper, we introduced controlled discrete-time semi-Markov random evolutions. These processes are random evolutions of discrete-time semi-Markov processes where we consider a control. applied to the values of random evolution. The main results concern time-rescaled weak convergence limit theorems in a Banach space of the above stochastic systems as averaging and diffusion approximation. The applications are given to the controlled additive functionals, controlled geometric Markov renewal processes, and controlled dynamical systems. We provide dynamical principles for discrete-time dynamical systems such as controlled additive functionals and controlled geometric Markov renewal processes. We also produce dynamic programming equations (Hamilton–Jacobi–Bellman equations) for the limiting processes in diffusion approximation such as controlled additive functionals, controlled geometric Markov renewal processes and controlled dynamical systems. As an example, we consider the solution of portfolio optimization problem by Merton for the limiting controlled geometric Markov renewal processes in diffusion approximation scheme. The rates of convergence in the limit theorems are also presented.

Central limit theorems for multivariate semi-Markov sequences and processes, with applications

1999 ◽  
Vol 36 (02) ◽  
pp. 415-432 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Central Limit ◽  
Limit Theorems ◽  
Markov Models ◽  
Central Limit Theorems ◽  
Renewal Processes ◽  
Counting Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Processes

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

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