Expectation of the Largest Betting Size in Labouchère System

07/31/2018

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by   Yanjun Han, et al.

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For Labouchère system with winning probability p at each coup, we prove that the expectation of the largest betting size under any initial list is finite if p>1/2, and is infinite if p<1/2, solving the open conjecture in grimmett2001one. The same result holds for a general family of betting systems, and the proof builds upon a recursive representation of the optimal betting system in the larger family.