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Doob, Ignatov and optional skipping. (English) Zbl 1040.60034
Let \(\xi_1,\xi_2,\dots\) be an i.i.d. sequence of random variables, taking values in a measurable space with common distribution \(F\). The authors describe a general set of conditions on a sequence of stopping times \(\tau_1,\tau_2,\dots\), not depending on \(F\), which guarantee that \(\xi_{\tau_1},\xi_{\tau_2}.\dots\) be i.i.d. with common distribution \(F\). This work is motivated by theorems of J. L. Doob [Ann. Math. (2) 37, 363–367 (1936; Zbl 0013.40802)] and Z. Ignatov [God. Sofij. Univ., Fak. Mat. Mekh. 71 (1976/1977), Part II, 79–94 (1986; Zbl 0619.60052)]. The authors unify and extend aspects of both.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
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[12] CHAPEL HILL, NORTH CAROLINA 27599-3260 E-MAIL: simons@stat.unc.edu Y.-C. YAO INSTITUTE OF STATISTICAL SCIENCE ACADEMIA SINICA TAIPEI TAIWAN E-MAIL: yao@stat.sinica.edu.tw L. YANG DEPARTMENT OF STATISTICS MICHIGAN STATE UNIVERSITY EAST LANSING, MICHIGAN 48824 E-MAIL: yang@stt.msu.edu
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