Patterns in Random Walks and Brownian Motion

Abstract

We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

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