Abstract
In this paper, first, we prove some inequalities for randomly stopped variables, which arise naturally in the gambling theory, then we show that a theorem of Chacon and some pointwise convergence theorems, which imply the submartingale convergence theorem, are immediate consequences of these inequalities.
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Chen, R. Some inequalities for randomly stopped variables with applications to pointwise convergence. Z. Wahrscheinlichkeitstheorie verw Gebiete 36, 75–83 (1976). https://doi.org/10.1007/BF00533210
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DOI: https://doi.org/10.1007/BF00533210