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.
Article PDF
Similar content being viewed by others
References
Chung, K.L., Erdős, P.: On the application of the Borel-Cantelli lemma. Trans. Amer. Math. Soc. 72, 179–186 (1952)
Chung, K.L., Hunt, G.A.: On the zeros of \(\mathop \sum \limits_1^n \pm 1\). Ann. of Math. 50, 385–400 (1949)
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)
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)
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)
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)
Knight, F.B.: Essentials of Brownian motion and diffusion. Am. Math. Soc. Providence, Rhode Island, 1981
Shepp, L.A.: A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912–1914 (1967)
Spitzer, F.: Principles of random walk. Princeton, N.J.: Van Nostrand, 1964
Steinebach, J.: A strong law of Erdős-Rényi type for cumulative processes in renewal theory. J. Appl. Probability 15, 96–111 (1978)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Csáki, E., Erdős, P. & Révész, P. On the length of the longest excursion. Z. Wahrscheinlichkeitstheorie verw Gebiete 68, 365–382 (1985). https://doi.org/10.1007/BF00532646
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00532646