Abstract
Using a martingale technique we derive bounds and asymptotics for tail distributions of first passage times \(\tau _{b}\) associated with crossing a level b for some gambling strategies. In particilar, for the case of martingale games with so-called “Oscar strategy” we show that \(P\{\tau _{b}>n\}\le C n^{-3/2}\,\) for any level \(b>0\).
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Notes
\(I\{A\}\) is an indicator function of an event A.
Here and further C denotes any positive constant not depending on b and n.
C denotes any generic positive constant not depending on the parameters n, \(\lambda \).
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Research of A. Novikov was partially supported by ARC Discovery Grant DP150102758.
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Novikov, A., Kaji, S. On distibutions of first passage times of martingales arising in some gambling problems. Japan J. Indust. Appl. Math. 34, 859–871 (2017). https://doi.org/10.1007/s13160-017-0269-5
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DOI: https://doi.org/10.1007/s13160-017-0269-5