Hostname: page-component-7dd5485656-bt4hw Total loading time: 0 Render date: 2025-10-31T04:33:00.629Z Has data issue: false hasContentIssue false
  • English
  • Français

Renewal theorems for processes with dependent interarrival times

Part of: Special processes Classical measure theory

Published online by Cambridge University Press:  29 November 2018

Sabrina Kombrink
Sabrina Kombrink*
Affiliation:
Universität zu Lübeck and Georg-August-Universität Göttingen
*
* Postal address: Mathematical Institute, Georg-August-Universität Göttingen, Bunsenstrasse 3-5, 37073 Göttingen, Germany. Email address: sabrina.kombrink@mathematik.uni-goettingen.de
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Renewal theoremdependent interarrival timesymbolic dynamicskey renewal theoremRuelle‒Perron‒Frobenius theory

MSC classification

Primary: 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes
Secondary: 28A80: Fractals 28A75: Length, area, volume, other geometric measure theory

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1193 - 1216
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alsmeyer, G. (1991).Erneuerungstheorie [Renewal Theory].B. G. Teubner,Stuttgart (in German).Google Scholar
[2]Asmussen, S. (2003).Applied Probability and Queues (Appl. Math. (New York) 51),2nd edn.Springer,New York.Google Scholar
[3]Bedford, T. (1988).Hausdorff dimension and box dimension in self-similar sets. In Proceedings of the Conference on Topology and Measure V, Ernst-Moritz-Arndt Universitat, Greifswald, pp. 17–26.Google Scholar
[4]Berry, M. V. (1979). Distribution of modes in fractal resonators. In Structural Stability in Physics (Springer Ser. Synergetics 4; Tübingen, 1978),Springer,Berlin, pp. 51–53.Google Scholar
[5]Berry, M. V. (1980).Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals. In Geometry of the Laplace Operator (Proc. Symp. Pure Math. XXXVI; Hawaii, 1979).American Mathematical Society,Providence, RI, pp. 13–28.Google Scholar
[6]Bowen, R. (2008).Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes Math. 470),2nd revised edn.Springer,Berlin.Google Scholar
[7]Falconer, K. J. (1995).On the Minkowski measurability of fractals.Proc. Amer. Math. Soc. 123,1115–1124.Google Scholar
[8]Falconer, K. (2003).Fractal Geometry. Mathematical Foundations and Applications,2nd edn.John Wiley,Hoboken, NJ.Google Scholar
[9]Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.Google Scholar
[10]Gatzouras, D. (2000).Lacunarity of self-similar and stochastically self-similar sets.Trans. Amer. Math. Soc. 352,1953–1983.Google Scholar
[11]Hutchinson, J. E. (1981).Fractals and self-similarity.Indiana Univ. Math. J. 30,713–747.Google Scholar
[12]Kato, T. (1995).Perturbation Theory for Linear Operators.Springer,Berlin.Google Scholar
[13]Kesseböhmer, M. and Kombrink, S. (2012).Fractal curvature measures and Minkowski content for self-conformal subsets of the real line.Adv. Math. 230,2474–2512.Google Scholar
[14]Kesseböhmer, M. and Kombrink, S. (2015).Minkowski content and fractal Euler characteristic for conformal graph directed systems.J. Fractal Geom. 2,171–227.Google Scholar
[15]Kesseböhmer, M. and Kombrink, S. (2017).A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory.Discrete Contin. Dynam. Systems Ser. S 10,335–352.Google Scholar
[16]Kesseböhmer, M. and Kombrink, S. (2017). Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings. Preprint. Available at https://arxiv.org/abs/1702.02854v1.Google Scholar
[17]Klenke, A. (2008).Wahrscheinlichkeitstheorie,2nd revised edn.Springer,Berlin.Google Scholar
[18]Kombrink, S. (2013).A survey on Minkowski measurability of self-similar and self-conformal fractals in ℝd. In Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. I. Fractals in Pure Mathematics (Contemp. Math. 600).American Mathematical Society,Providence, RI, pp. 135–159.Google Scholar
[19]Lalley, S. P. (1989).Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits.Acta Math. 163,1–55.Google Scholar
[20]Lapidus, M. L. and Pomerance, C. (1993).The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums.Proc. London Math. Soc. 66,41–69.Google Scholar
[21]Lapidus, M. L. and van Frankenhuijsen, M. (2006).Fractal Geometry, Complex Dimensions and Zeta Functions. Geometry and Spectra of Fractal Strings.Springer,New York.Google Scholar
[22]Mauldin, R. D. and Urbański, M. (1996).Dimensions and measures in infinite iterated function systems.Proc. London Math. Soc. 73,105–154.Google Scholar
[23]Melbourne, I. and Terhesiu, D. (2012).Operator renewal theory and mixing rates for dynamical systems with infinite measure.Invent. Math. 189,61–110. (Correction:202 (2015),1269–1272.)Google Scholar
[24]Parry, W. and Pollicott, M. (1990).Zeta functions and the periodic orbit structure of hyperbolic dynamics.Astérisque 187‒188, 268pp.Google Scholar
[25]Pollicott, M. and Urbański, M. (2017). Asymptotic counting in conformal dynamical systems. Preprint. Available at https://arxiv.org/abs/1704.06896.Google Scholar
[26]Pollicott, M. (1984).A complex Ruelle-Perron-Frobenius theorem and two counterexamples.Ergodic Theory Dynam. Systems 4,135–146.Google Scholar
[27]Ruelle, D. (2004).Thermodynamic Formalism,2nd edn.Cambridge University Press.Google Scholar
[28]Sarig, O. (2002).Subexponential decay of correlations.Invent. Math. 150,629–653.Google Scholar
[29]Walters, P. (1982).An Introduction to Ergodic Theory (Graduate Texts Math. 79).Springer,New York.Google Scholar
[30]Walters, P. (2001).Convergence of the Ruelle operator for a function satisfying Bowen's condition.Trans. Amer. Math. Soc. 353,327–347.Google Scholar
[31]Winter, S. (2015).Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets.Adv. Math. 274,285–322.Google Scholar