Periodicity in Markov renewal theory

1974 ◽  
Vol 6 (1) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Regenerative Processes ◽  
Markov Kernel ◽  
Transient States ◽  
Initial States ◽  
Markov Renewal Theory ◽  
Direct Inspection

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

Periodicity in Markov renewal theory

1974 ◽  
Vol 6 (01) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Regenerative Processes ◽  
Markov Kernel ◽  
Transient States ◽  
Initial States ◽  
Markov Renewal Theory ◽  
Direct Inspection

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

scholarly journals Theory of semiregenerative phenomena

1988 ◽  
Vol 25 (A) ◽  
pp. 257-274
Author(s):  
N. U. Prabhu
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Renewal Process ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Discrete Time Case ◽  
Markov Renewal Theory

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

scholarly journals Theory of semiregenerative phenomena

1988 ◽  
Vol 25 (A) ◽  
pp. 257-274 ◽  
Author(s):  
N. U. Prabhu
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Renewal Process ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Discrete Time Case ◽  
Markov Renewal Theory

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

Regenerative processes in the infinite mean cycle case

2001 ◽  
Vol 38 (01) ◽  
pp. 165-179 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Regenerative Process ◽  
Distribution Functions ◽  
Regenerative Processes ◽  
Alternating Renewal Process ◽  
Different Types ◽  
Mean Cycle ◽  
Regenerative Cycle

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

Regenerative processes in the infinite mean cycle case

2001 ◽  
Vol 38 (1) ◽  
pp. 165-179 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Regenerative Process ◽  
Distribution Functions ◽  
Regenerative Processes ◽  
Alternating Renewal Process ◽  
Different Types ◽  
Mean Cycle ◽  
Regenerative Cycle

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (3) ◽  
pp. 614-620 ◽  
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn), n ≧ 0} be the standard J–X process of Markov renewal theory. Suppose {Jn, n ≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that if converges in distribution, where an, bn > 0 (bn → ∞) are real constants, then the limit law F must be stable. Suppose Q(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn), n ≧ 0}. Then the n-fold convolution, Q∗n(bnx + anbn), converges in distribution to F(x)Π if and only if converges in distribution to F. Π is the matrix of stationary transition probabilities of {Jn, n ≧ 0}. Sufficient conditions on the Hi's are given for the convergence of the sequence of semi-Markov matrices to F(x)Π, where F is stable.

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

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