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2008 Classes of measures which can be embedded in the Simple Symmetric Random Walk
Alexander Cox, Jan Obloj
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Electron. J. Probab. 13: 1203-1228 (2008). DOI: 10.1214/EJP.v13-516


We characterize the possible distributions of a stopped simple symmetric random walk $X_\tau$, where $\tau$ is a stopping time relative to the natural filtration of $(X_n)$. We prove that any probability measure on $\mathbb{Z}$ can be achieved as the law of $X_\tau$ where $\tau$ is a minimal stopping time, but the set of measures obtained under the further assumption that $(X_{n\land \tau}:n\geq 0)$ is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on $\mathbb{Z}$. This is in sharp contrast to the well-studied Brownian motion setting. We also investigate the discrete counterparts of the Chacon-Walsh (1976) and Azema-Yor (1979) embeddings and show that they lead to yet smaller sets of achievable measures. Finally, we solve explicitly the Skorokhod embedding problem constructing, for a given measure $\mu$, a minimal stopping time $\tau$ which embeds $\mu$ and which further is uniformly integrable whenever a uniformly integrable embedding of $\mu$ exists.


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Alexander Cox. Jan Obloj. "Classes of measures which can be embedded in the Simple Symmetric Random Walk." Electron. J. Probab. 13 1203 - 1228, 2008.


Accepted: 31 July 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1196.60080
MathSciNet: MR2430705
Digital Object Identifier: 10.1214/EJP.v13-516

Primary: 60G50
Secondary: 28A80 , 60G40

Keywords: Azema-Yor stopping time , Chacon-Walsh stopping time , Fractal , iterated function system , minimal stopping time , Random walk , self-similar set , Skorokhod embedding problem , uniform integrability

Vol.13 • 2008
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