Electronic Journal of Probability

The argmin process of random walks, Brownian motion and Lévy processes

Jim Pitman and Wenpin Tang

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Abstract

In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha _t:=\sup \left \{s \in [0,1]: B_{t+s}=\inf _{u \in [0,1]}B_{t+u} \right \}$ for $t \geq 0$. The argmin process $\alpha $ is stationary, with invariant measure which is arcsine distributed. We prove that $(\alpha _t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot )$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed renewal process with an explicit path construction. We also give a path decomposition for Brownian motion at these extrema.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 60, 35 pp.

Dates
Received: 22 August 2017
Accepted: 6 June 2018
First available in Project Euclid: 20 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1529460158

Digital Object Identifier
doi:10.1214/18-EJP185

Mathematical Reviews number (MathSciNet)
MR3827967

Zentralblatt MATH identifier
06924672

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; Lévy processes 60J65: Brownian motion [See also 58J65]

Keywords
arcsine law argmin process Brownian extrema Feller semigroup Brownian excursion theory jump process Lévy process Lévy system Markov property space-time shift process path decomposition random walks renewal property sample path property stable process stationary process

Rights
Creative Commons Attribution 4.0 International License.

Citation

Pitman, Jim; Tang, Wenpin. The argmin process of random walks, Brownian motion and Lévy processes. Electron. J. Probab. 23 (2018), paper no. 60, 35 pp. doi:10.1214/18-EJP185. https://projecteuclid.org/euclid.ejp/1529460158


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