On the length of the longest excursion

  • E. Csáki
  • P. Erdős
  • P. Révész
Article

Summary

A lower limit of the length of the longest excursion of a symmetric random walk is given. Certain related problems are also discussed. It is shown e.g. that for any ɛ>0 and all sufficiently large n there are c(ɛ) loglog n excursions in the interval (0, n) with total length greater than n(1−ɛ), with probability 1.

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References

  1. Chung, K.L., Erdős, P.: On the application of the Borel-Cantelli lemma. Trans. Amer. Math. Soc. 72, 179–186 (1952)Google Scholar
  2. Chung, K.L., Hunt, G.A.: On the zeros of \(\mathop \sum \limits_1^n \pm 1\). Ann. of Math. 50, 385–400 (1949)Google Scholar
  3. Csáki, E., Csörgő, M., Földes, A., Révész, P.: How big are the increments of the local time of a Wiener process. Ann. Probability 11, 593–608 (1983)Google Scholar
  4. Csáki, E., Földes, A.: How big are the increments of the local time of a simple symmetric random walk? Coll. Math. Soc. J. Bolyai 36. Limit theorems in probability and statistics,Veszprém (Hungary), 1982. P. Révész (ed.). (To appear)Google Scholar
  5. Csáki, E., Révész, P.: A combinatorial proof of a theorem of P. Lévy on the local time. Acta Sci. Math. (Szeged) 45, 119–129 (1983)Google Scholar
  6. Greenwood, P., Perkins, E.: A conditioned limit theorem for random walk and Brownian local time on square root boundaries. Ann. Probability 11, 227–261 (1983)Google Scholar
  7. Knight, F.B.: Essentials of Brownian motion and diffusion. Am. Math. Soc. Providence, Rhode Island, 1981Google Scholar
  8. Shepp, L.A.: A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912–1914 (1967)Google Scholar
  9. Spitzer, F.: Principles of random walk. Princeton, N.J.: Van Nostrand, 1964Google Scholar
  10. Steinebach, J.: A strong law of Erdős-Rényi type for cumulative processes in renewal theory. J. Appl. Probability 15, 96–111 (1978)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • E. Csáki
    • 1
  • P. Erdős
    • 1
  • P. Révész
    • 1
  1. 1.Mathematical InstituteBudapestHungary

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