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The most visited site of Brownian motion and simple random walk

  • Richard F. Bass
  • Philip S. Griffin
Article

Summary

Let L(t, x) be the local time at x for Brownian motion and for each t, let \(\bar V(t) = \inf \{ x\underline{\underline > } 0;L(t,x) \vee L(t, - x) = \mathop {\sup }\limits_y L(t,y)\} \), the absolute value of the most visited site for Brownian motion up to time t. In this paper we prove that ¯V(t) is transient and obtain upper and lower bounds for the rate of growth of ¯V(t). The main tools used are the Ray-Knight theorems and William's path decomposition of a diffusion. An invariance principle is used to get analogous results for simple random walks. We also obtain a law of the iterated logarithm for ¯V(t).

Keywords

Lower Bound Stochastic Process Brownian Motion Random Walk Probability Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Richard F. Bass
    • 1
  • Philip S. Griffin
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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