Summary
Let N(t) be the local time at zero (the number of returns to zero up to time t) of a recurrent random walk. Consider the largest increments over subintervals of length a t
The almost sure behaviour of \(\tilde Y(t)\)is shown to be the same as the behaviour of the corresponding increments of the local time of a Wiener process provided a t/log t→∞. In the case a t=c log t an Erdős-Rényi type result is obtained.
Article PDF
Similar content being viewed by others
References
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). To appear 1982
Csáki, E., Révész, P.: Strong invariance for local times. Z. Wahrscheinlichkeitstheorie verw. Gebiete 62, 263–278 (1983)
Révész, P.: Local time and invariance, Analytical Methods in Probability Theory. (Proceedings, Oberwolfach, Germany, 1980), Ed. Dugue, Lukács, Rohatgi. Lecture Notes in Math. 861, 128–145. Berlin-Heidelberg-New York: Springer 1981
Spitzer, F.: Principles of random walk. Princeton: 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)
Steinebach, J.: Between invariance principles and Erdős-Rényi laws. Coll. Math. Soc. J. Bolyai 36. Limit theorems in probability and statistics, Veszprém (Hungary) 1982 (to appear)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Csáki, E., Földes, A. How big are the increments of the local time of a recurrent random walk?. Z. Wahrscheinlichkeitstheorie verw Gebiete 65, 307–322 (1983). https://doi.org/10.1007/BF00532485
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00532485