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Classes of measures which can be embedded in the simple symmetric random walk. (English) Zbl 1196.60080

Summary: 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 Z can be achieved as the law of X stopped at a minimal stopping time, but the set of measures obtained under the further assumption that stopped process is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on Z. This is in sharp contrast to the well-studied Brownian motion setting. We also investigate the discrete counterparts of the R. V. Chacon and J. B. Walsh [Semin. Probab. X, Univ. Strasbourg 1974/75, Lect. Notes Math. 511, 19–23 (1976; Zbl 0329.60041)] and J. Azéma and M. Yor [Seminaire de probabilites XIII, Univ. Strasbourg 1977/78, Lect. Notes Math. 721, 90–115 (1979; Zbl 0414.60055)] 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.

MSC:

60G50 Sums of independent random variables; random walks
60G40 Stopping times; optimal stopping problems; gambling theory
28A80 Fractals
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