Gambling Systems and Multiplication-Invariant Measures
نویسندگان
چکیده
This short paper describes a surprising connection between two previously unrelated topics: the probability of winning certain gambling games, and the invariance of certain measures under pointwise multiplication on the circle. The former has been well-studied by probabilists, notably in the book of Dubins and Savage (1965). The latter is the subject of an important and well-studied conjecture of H. Furstenberg. But to the best of our knowledge, the connection between them is new. We shall show that associated with the gambling games are certain measures (whose cumulative distribution function values F (x) are equal to the probability of winning the gambling game when starting with initial fortune x and using the strategy of “Bold Play”). We shall then show that these measures have interesting properties related to multiplication invariance; in particular, they give rise to a collection of measures which are invariant under multiplication by 2, and which have a weak limit which is also invariant under multiplication by 3. These measures provide some candidates for possible counterexamples to Furstenberg’s conjecture. Necessary background about gambling is presented in Section 2. Necessary background about multiplication-invariant measures is presented in Section 3. The connection between the two is discussed in Section 4. Some further observations are in Section 5.
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