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Limit Theorems for Local and Occupation Times of Random Walks and Brownian Motion on a Spider

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Abstract

A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs, we establish limit theorems for n-step local and occupation times.

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References

  1. Appuhamillage, T.A., Bokil, V., Thomann, E., Waymire, E., Wood, B.: Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann. Appl. Probab. 21, 183–214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barlow, M.T., Pitman J.W., Yor M.: Une extension multidimensionelle de la loi de l’arc sinus. Sém. Prob. XXIII, Lect.Notes in Math.,vol. 1372. Springer, Berlin, Heidelberg, New York, pp. 294–314 (1989)

  3. Bass, R.F., Griffin, P.S.: The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verwandte Geb. 70, 417–436 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel (1996)

    Book  MATH  Google Scholar 

  5. Burdzy, K., Chen, Z.-Q.: Local time flow related to skew Brownian motion. Ann. Probab. 29, 1693–1715 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cherny, A., Shiryaev, A., Yor, M.: Limit behavior of the “horizontal–vertical” random walk and some extension of the Donsker–Prokhorov invariance principle. Theory Probab. Appl. 47, 377–394 (2004)

    Article  MATH  Google Scholar 

  7. Chung, K.L., Erdős, P.: On the application of the Borel–Cantelli lemma. Trans. Am. Math. Soc. 64, 179–186 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  8. Csáki, E., Csörgő, M., Földes, A., Révész, P.: How big are the increments of the local time of a Wiener process? Ann. Probab. 11, 593–608 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Csáki, E., Csörgő, M., Földes, A., Révész, P.: Strong approximation of additive functionals. J. Theory Probab. 5, 679–706 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Csáki, E., Csörgő, M., Földes, A., Révész, P.: Global Strassen-type theorems for iterated Brownian motions. Stoch. Process. Appl. 59, 321–341 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Csáki, E., Csörgő, M., Földes, A., Révész, P.: Some limit theorems for heights of random walks on a spider. J. Theor. Probab. 29, 1685–1709 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Csáki, E., Földes, A., Révész, P.: Strassen theorems for a class of iterated processes. Trans. Am. Math. Soc. 349, 1153–1167 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Csáki, E., Salminen, P.: On additive functionals of diffusion processes. Stud. Sci. Math. Hung. 31, 47–62 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Csörgő, M., Horváth, L.: On best possible approximations of local time. Stat. Probab. Lett. 8, 301–306 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dobrushin, R.L.: Two limit theorems for the simplest random walk on a line. Usp. Mat. Nauk (N.S.) 10(3(65)), 139–146 (1955). (in Russian)

  16. Evans, S.N.: Snakes and spiders: Brownian motion on \(\mathbb{R}\)-trees. Probab. Theory Relat. Fields 117, 361–386 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gairat, A., Shcherbakov, V.: Density of skew Brownian motion and its functionals with application in finance. Math. Finance 27, 1069–1088 (2017). doi:10.1111/mafi.12120

    Article  MathSciNet  MATH  Google Scholar 

  18. Hajri, H.: Discrete approximations to solution flows of Tanaka’s SDE related to Walsh Brownian motion. Sém. Prob. XLIV, Lect. Notes in Math., vol. 2046. Springer, Berlin, Heidelberg, New York, pp. 167–190 (2012)

  19. Harrison, J., Shepp, L.: On skew Brownian motion. Ann. Probab. 9, 309–313 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hu, Y., Pierre-Loti-Viaud, D., Shi, Z.: Laws of the iterated logarithm for iterated Wiener processes. J. Theor. Probab. 8, 303–319 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Itô, K., McKean, H.: Diffusion and Their Sample Paths, 2nd edn. Springer, Berlin (1974)

    MATH  Google Scholar 

  22. Kesten, H.: An iterated logarithm law for the local time. Duke Math. J. 32, 447–456 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  23. Komlós, M., Major, P., Tusnády, G.: An approximation of partial sums of independent r.v’.s and sample df. I. Z. Wahrsch. Verwandte Geb. 32, 111–131 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lamperti, J.: An occupation time theorem for a class of stochastic processes. Trans. Am. Math. Soc. 88, 380–387 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lejay, A.: On the constructions of the skew Brownian motion. Probab. Surv. 3, 413–466 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lejay, A.: Estimation of the bias parameter of the Skew Random Walk and application to the Skew Brownian Motion. Stat. Inference Stoch. Process (2017). doi:10.1007/s11203-017-9161-9

  27. Lyulko, Y.A.: On the distribution of time spent by a Markov chain at different levels until achieving a fixed state. Theory Probab. Appl. 56, 140–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Petrov, V.V.: Sums of Independent Random Variables. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  29. Révész, P.: Random Walk in Random and Non-random Environment, 3d edn. World Scientific, Singapore (2013)

    Book  MATH  Google Scholar 

  30. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  31. Spitzer, F.: Principles of Random Walk. Van Nostrand, Princeton (1964)

    Book  MATH  Google Scholar 

  32. Walsh, J.B.: A diffusion with discontinuous local time. Astérisque 52–53, 37–45 (1978)

    Google Scholar 

  33. Watanabe, S.: Generalized arc-sine laws for one-dimensional diffusion processes and random walks. In: Proceedings of Symposia in Pure Mathematics, vol. 57. Stoch. Analysis, Cornell University (1993), Am. Math. Soc., pp. 157–172 (1995)

  34. Yano, Y.: On the joint law of the occupation times for a diffusion process on multiray. J. Theor. Probab. 30, 490–509 (2017). doi:10.1007/s10959-015-0654-4

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We sincerely wish to thank the referee of our submission for careful reading our manuscript, and for making a number of insightful comments and suggestions that helped and prompted us to improve the presentation and proofs of our results when revising this paper for publication.

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O.B. 127, Budapest, 1364, Hungary

    Endre Csáki

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Miklós Csörgő

  3. Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, NY, 10314, USA

    Antónia Földes

  4. Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/107, Vienna, 1040, Austria

    Pál Révész

Authors
  1. Endre Csáki
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  2. Miklós Csörgő
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  3. Antónia Földes
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  4. Pál Révész
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Corresponding author

Correspondence to Endre Csáki.

Additional information

Endre Csáki research supported by Hungarian National Research, Development and Innovation Office-NKFIH Grant No. K 108615. Miklós Csörgő research supported by an NSERC Canada Discovery Grant at Carleton University. Antónia Földes research supported by a PSC CUNY Grant No. 69040-0047. Pál Révész research supported by Hungarian National Research, Development and Innovation Office-NKFIH Grant No. K 108615.

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Csáki, E., Csörgő, M., Földes, A. et al. Limit Theorems for Local and Occupation Times of Random Walks and Brownian Motion on a Spider. J Theor Probab 32, 330–352 (2019). https://doi.org/10.1007/s10959-017-0788-7

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  • DOI: https://doi.org/10.1007/s10959-017-0788-7

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