Laplace Transform for the Key Renewal Theorem

2005 ◽  
pp. 359-360
Keyword(s):  
Laplace Transform ◽  
Renewal Theorem ◽  
Key Renewal Theorem

Applications of the key renewal theorem: crudely regenerative processes

1992 ◽  
Vol 29 (02) ◽  
pp. 384-395
Author(s):  
Richard F. Serfozo
Keyword(s):  
Classical Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Function ◽  
Regenerative Processes ◽  
Renewal Theorem ◽  
Key Renewal Theorem ◽  
Derivatives Of

Limit Statements obtainable by the key renewal theorem are of the form EXt = v(t) + o(1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EXt.

scholarly journals Superposition of renewal processes

1991 ◽  
Vol 23 (01) ◽  
pp. 64-85 ◽  
Author(s):  
C. Y. Teresalam ◽  
John P. Lehoczky
Keyword(s):  
Asymptotic Behavior ◽  
Queueing Systems ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Key Renewal Theorem

This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of queueing systems.

Uniform limit theorems for non-singular renewal and Markov renewal processes

1978 ◽  
Vol 15 (1) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie
Keyword(s):  
Recurrence Time ◽  
Renewal Processes ◽  
Convergence Properties ◽  
Renewal Theorem ◽  
Uniform Limit ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Positive Recurrent ◽  
Forward Recurrence Time ◽  
Initial Distributions

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

Applications of the key renewal theorem: crudely regenerative processes

1992 ◽  
Vol 29 (2) ◽  
pp. 384-395 ◽  
Author(s):  
Richard F. Serfozo
Keyword(s):  
Classical Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Function ◽  
Regenerative Processes ◽  
Renewal Theorem ◽  
Key Renewal Theorem ◽  
Derivatives Of

Limit Statements obtainable by the key renewal theorem are of the form EXt = v(t) + o(1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EXt.

Renewal theorems for processes with dependent interarrival times

2018 ◽  
Vol 50 (4) ◽  
pp. 1193-1216
Author(s):  
Sabrina Kombrink
Keyword(s):  
Point Processes ◽  
Renewal Theory ◽  
Renewal Theorem ◽  
Hölder Continuous ◽  
Finite State ◽  
Key Renewal Theorem ◽  
Interarrival Times ◽  
Markov Renewal Theory ◽  
Discrete Measures ◽  
Finite State Space

Abstract In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry, for instance to the problem of Minkowski measurability of self-conformal sets.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (2) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

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