scholarly journals A martingale view of Blackwell’s renewal theorem and its extensions to a general counting process

2019 ◽  
Vol 56 (2) ◽  
pp. 602-623
Author(s):  
Daryl J. Daley ◽  
Masakiyo Miyazawa
Keyword(s):  
Asymptotic Behaviour ◽  
Renewal Process ◽  
Stochastic Analysis ◽  
Counting Process ◽  
Intensity Function ◽  
Counting Processes ◽  
Renewal Theorem ◽  
Basic Tool ◽  
Semimartingale Representation ◽  
Blackwell's Renewal Theorem

AbstractMartingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell’s renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell’s renewal theorem holds.

An age-dependent counting process generated from a renewal process

1988 ◽  
Vol 20 (4) ◽  
pp. 739-755 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Renewal Process ◽  
Counting Process ◽  
Intensity Function ◽  
Sufficient Condition ◽  
Minimal Repair ◽  
Homogeneous Poisson Process ◽  
Stochastic Convexity ◽  
Age Dependent ◽  
Non Homogeneous Poisson Process ◽  
Age Process

Let {N(t)} be a renewal process having the associated age process {X(t)}. Of interest is the counting process {M(t)} characterized by a non-homogeneous Poisson process with age-dependent intensity function λ (X(t)). The trivariate process {Y(t) = [M(t), N(t), X(t)]} is analyzed obtaining its Laplace transform generating function explicitly. Based on this result, asymptotic behavior of {S(t) = cM(t) + dN(t)} as t → ∞ is discussed. Furthermore, a sufficient condition is given under which {M(t), –N(t), X(t)} is stochastically monotone and associated. This condition also assures increasing stochastic convexity of {M(t)}. The usefulness of these results is demonstrated through an application to the age-dependent minimal repair problem.

An age-dependent counting process generated from a renewal process

1988 ◽  
Vol 20 (04) ◽  
pp. 739-755 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Renewal Process ◽  
Counting Process ◽  
Intensity Function ◽  
Sufficient Condition ◽  
Minimal Repair ◽  
Homogeneous Poisson Process ◽  
Stochastic Convexity ◽  
Age Dependent ◽  
Non Homogeneous Poisson Process ◽  
Age Process

Let {N(t)} be a renewal process having the associated age process {X(t)}. Of interest is the counting process {M(t)} characterized by a non-homogeneous Poisson process with age-dependent intensity function λ (X(t)). The trivariate process {Y(t) = [M(t), N(t), X(t)]} is analyzed obtaining its Laplace transform generating function explicitly. Based on this result, asymptotic behavior of {S(t) = cM(t) + dN(t)} as t → ∞ is discussed. Furthermore, a sufficient condition is given under which {M(t), –N(t), X(t)} is stochastically monotone and associated. This condition also assures increasing stochastic convexity of {M(t)}. The usefulness of these results is demonstrated through an application to the age-dependent minimal repair problem.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

A note on filtered Markov renewal processes

1981 ◽  
Vol 18 (03) ◽  
pp. 752-756
Author(s):  
Per Kragh Andersen
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
The Moment

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)

1996 ◽  
Vol 28 (03) ◽  
pp. 853-876 ◽  
Author(s):  
Philippe Picard ◽  
Claude Lefèvre
Keyword(s):  
General Framework ◽  
Counting Process ◽  
Lower Boundary ◽  
Exact Distribution ◽  
Applied Probability ◽  
Renewal Processes ◽  
Counting Processes ◽  
Unified Approach ◽  
Non Linear ◽  
Birth Processes

The paper is concerned with the distribution of the levelNof the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution ofNrequires the use of a classical family of Abel–Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.

RENEWAL PROCESS FOR FUZZY VARIABLES

2007 ◽  
Vol 15 (04) ◽  
pp. 493-501 ◽  
Author(s):  
DUG HUN HONG
Keyword(s):  
Renewal Process ◽  
Renewal Theorem ◽  
Arrival Times ◽  
Fuzzy Variables ◽  
Continuity Assumption

Recently, Zhao and Liu [IJUFKS 11 (2003) 573–586] proposed a "fuzzy elementary renewal theorem" and "fuzzy renewal rewards theorem" for a renewal process in which the inter-arrival times and rewards are characterized as continuous fuzzy variables. The continuity assumption is restrictive. In this note, we prove the same results without the assumption of continuity of the inter-arrival times and rewards.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (04) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂ t , the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂ t , and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (4) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂t, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂t, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

scholarly journals Delayed Renewal Process with Uncertain Random Inter-Arrival Times

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1943
Author(s):  
Xiaoli Wang ◽  
Gang Shi ◽  
Yuhong Sheng
Keyword(s):  
Random Process ◽  
Renewal Process ◽  
Random Variable ◽  
Random Delay ◽  
Renewal Theorem ◽  
Arrival Times ◽  
Chance Distribution ◽  
Renewal Rate ◽  
First Arrival

An uncertain random variable is a tool used to research indeterminacy quantities involving randomness and uncertainty. The concepts of an ’uncertain random process’ and an ’uncertain random renewal process’ have been proposed in order to model the evolution of an uncertain random phenomena. This paper designs a new uncertain random process, called the uncertain random delayed renewal process. It is a special type of uncertain random renewal process, in which the first arrival interval is different from the subsequent arrival interval. We discuss the chance distribution of the uncertain random delayed renewal process. Furthermore, an uncertain random delay renewal theorem is derived, and the chance distribution limit of long-term expected renewal rate of the uncertain random delay renewal system is proved. Then its average uncertain random delay renewal rate is obtained, and it is proved that it is convergent in the chance distribution. Finally, we provide several examples to illustrate the consistency with the existing conclusions.

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